%------------------------------------------------------------------------------
% File     : ITP292_3 : TPTP v9.0.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Intf_Imperative 00161_004904
% 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    : 0096_VEBT_Intf_Imperative_00161_004904 [Des22]

% Status   : Theorem
% Rating   : 0.67 v8.1.0
% Syntax   : Number of formulae    : 13698 (2556 unt;3883 typ;   0 def)
%            Number of atoms       : 29413 (8800 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 21638 (2981   ~; 442   |;2021   &)
%                                         (2611 <=>;13583  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   16 (   2 avg)
%            Number of FOOLs       : 2447 ( 831 fml;1616 var)
%            Number of X terms     : 1099 (   0  []; 823 ite; 276 let)
%            Number of types       :  940 ( 939 usr)
%            Number of type conns  : 3992 (2458   >;1534   *;   0   +;   0  <<)
%            Number of predicates  :  216 ( 213 usr;   3 prp; 0-4 aty)
%            Number of functors    : 2755 (2755 usr; 509 con; 0-8 aty)
%            Number of variables   : 28482 (27224   !;1258   ?;28482   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 23:43:11.711
%------------------------------------------------------------------------------
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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__fun_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    fun_o_VEBT_VEBT: $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__fun_It__String__Oliteral_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__String__Oliteral_Mt__Int__Oint_J,type,
    fun_literal_int: $tType ).

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

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

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

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

tff(ty_n_t__fun_It__Nat__Onat_Mt__String__Oliteral_J,type,
    fun_nat_literal: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__String__Oliteral_J,type,
    fun_int_literal: $tType ).

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

tff(ty_n_t__fun_I_Eo_Mt__Code____Numeral__Ointeger_J,type,
    fun_o_Code_integer: $tType ).

tff(ty_n_t__fun_It__fun_It__Nat__Onat_M_Eo_J_M_Eo_J,type,
    fun_fun_nat_o_o: $tType ).

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

tff(ty_n_t__fun_It__Nat__Onat_Mt__Uint32__Ouint32_J,type,
    fun_nat_uint32: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Uint32__Ouint32_J,type,
    fun_int_uint32: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_o_fun_nat_o: $tType ).

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

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

tff(ty_n_t__Option__Ooption_It__Set__Oset_I_Eo_J_J,type,
    option_set_o: $tType ).

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

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

tff(ty_n_t__List__Olist_It__List__Olist_I_Eo_J_J,type,
    list_list_o: $tType ).

tff(ty_n_t__itself_It__Numeral____Type__Onum0_J,type,
    itself_Numeral_num0: $tType ).

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

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

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

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

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

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

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

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

tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__List__Olist_It__Assertions__Oassn_J,type,
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tff(ty_n_t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    heap_Time_Heap_o: $tType ).

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

tff(ty_n_t__List__Olist_It__String__Oliteral_J,type,
    list_literal: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__fun_It__List__Olist_I_Eo_J_M_Eo_J,type,
    fun_list_o_o: $tType ).

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

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

tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
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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,
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tff(ty_n_t__List__Olist_It__Uint32__Ouint32_J,type,
    list_uint32: $tType ).

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

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

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

tff(ty_n_t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
    fun_o_assn: $tType ).

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

tff(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    set_set_o: $tType ).

tff(ty_n_t__fun_It__String__Oliteral_M_Eo_J,type,
    fun_literal_o: $tType ).

tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_M_Eo_J,type,
    fun_set_o_o: $tType ).

tff(ty_n_t__Set__Oset_It__Uint32__Ouint32_J,type,
    set_uint32: $tType ).

tff(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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tff(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
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tff(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Uint32__Ouint32_M_Eo_J,type,
    fun_uint32_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
    fun_o_fun_o_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Uint32__Ouint32_J,type,
    fun_o_uint32: $tType ).

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

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

tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__Rat__Orat_J,type,
    list_rat: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(ty_n_t__Assertions__Oassn,type,
    assn: $tType ).

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

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

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

tff(ty_n_t__Uint32__Ouint32,type,
    uint32: $tType ).

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

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

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

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

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

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

% Explicit typings (2944)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_VEBT_VEBT > fun_VEBT_VEBT_o ).

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

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: fun_int_int ).

tff(sy_c_ATP_058Lamm__aaf____,type,
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tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_complex > fun_complex_o ).

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

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

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

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_VE5577048379444288083T_assn > fun_li1038900370375840338i_assn ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_VE1225457145571040039t_assn > fun_li8415865614262688720i_assn ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_VE2561185598974572172i_assn > fun_li6808080174506825141i_assn ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_VE5641251438918192003t_assn > fun_li762464189754283564i_assn ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_in7755481285845205148T_assn > fun_li2512619336610268315i_assn ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_int_fun_nat_assn > fun_li1069323510795985735i_assn ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_in1903134756389476099i_assn > fun_li80102209421453868i_assn ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_int_fun_int_assn > fun_li2639294123142356387i_assn ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_VE1571388752190420990T_assn > fun_li4421246802284715015i_assn ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_VE4931415397359990460t_assn > fun_li5819835350756573403i_assn ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_int_int ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_nat_o > fun_fu4001317426178549763_nat_o ).

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

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: vEBT_VEBT > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_nat_real ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_int_int ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: vEBT_VEBT > fun_fu8380412309343763075uint32 ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: real > fun_fu3826616023661743271uint32 ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: nat > fun_fu2798604719624125935uint32 ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: int > fun_fu8432976761865684647uint32 ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: complex > fun_fu3643814932998673835uint32 ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: code_integer > fun_fu2584227209315667977uint32 ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: vEBT_VEBT > fun_fu8380412309343763075uint32 ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: real > fun_fu3826616023661743271uint32 ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: nat > fun_fu2798604719624125935uint32 ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: int > fun_fu8432976761865684647uint32 ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: complex > fun_fu3643814932998673835uint32 ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: code_integer > fun_fu2584227209315667977uint32 ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_na6208021027755453088BT_nat ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_nat_fun_o_nat ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: nat > fun_complex_nat ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: nat > fun_Code_integer_nat ).

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

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: nat > fun_int_nat ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: nat > fun_o_nat ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_VE597728521808591204at_nat ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_nat_complex > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: vEBT_VEBT > fun_VE1035329831157012416o_assn ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: vEBT_VEBT > fun_VE7792148054706007876t_assn ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: vEBT_VEBT > fun_VE7792148054706007876t_assn ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_VEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: $o > fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: vEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_real > fun_real_o ).

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

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_complex_o > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_list_nat_o > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_set_nat_o > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_complex_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_nat_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_int_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_o_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_VEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: set_VEBT_VEBT > fun_fu1493198444879918881VEBT_o ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_real > fun_fu4757868175734369219real_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: set_VEBT_VEBT > fun_fu6979804496544055329VEBT_o ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: set_real > fun_fu8602227297639178947real_o ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: set_VEBT_VEBT > fun_fu1633589921238504865VEBT_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: set_real > fun_fu4472351043248340291real_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_VEBT_VEBT > fun_fu8552322614197228513VEBT_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_real > fun_fu3313596450186583811real_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_nat_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_nat_int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_nat_int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_int_nat > fun_fu1731011121079923219at_nat ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_int_nat > fun_fu3633881621402835147nt_nat ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_int_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_set_nat_nat > fun_fu1470602711428196973at_nat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_set_nat_int > fun_fu8203891603351246025at_int ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_VE3165209603783583043T_VEBT > fun_se3108574681309861106VEBT_o ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_VE3165209603783583043T_VEBT > fun_se3108574681309861106VEBT_o ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_real_VEBT_VEBT > fun_se6820275320689043114VEBT_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_real_VEBT_VEBT > fun_se5796227354453888814real_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_VEBT_VEBT_real > fun_se6982712487775864618real_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_VEBT_VEBT_real > fun_se3619762755820101550VEBT_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_real_real > fun_se1404052156289531106real_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_real_real > fun_se1404052156289531106real_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_co4340422964966740447T_VEBT > fun_se7882934732587789228VEBT_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_co4340422964966740447T_VEBT > fun_se6779386234326909994plex_o ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_complex_real > fun_se937755008525480164real_o ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_complex_real > fun_se7159489718065667682plex_o ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_nat_VEBT_VEBT > fun_se4740616104211847630VEBT_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_nat_VEBT_VEBT > fun_se7821215522990065638_nat_o ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_nat_real > fun_se4974600911086321158real_o ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_nat_real > fun_se7551511148262207202_nat_o ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_int_VEBT_VEBT > fun_se5185063861201754666VEBT_o ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_int_VEBT_VEBT > fun_se7866262565344360238_int_o ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_int_real > fun_se3448549103344627810real_o ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_int_real > fun_se6029058084618929634_int_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_uint32_uint32 ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_real_real ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: set_VEBT_VEBT > fun_fu8478390438599113818VEBT_o ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: set_real > fun_fu2964587086908134546real_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: set_VEBT_VEBT > fun_fu176909009941635838VEBT_o ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: set_real > fun_fu1867504106264135990real_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: set_VEBT_VEBT > fun_fu3260669801086416892VEBT_o ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: set_real > fun_fu5550611363604350772real_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: set_VEBT_VEBT > fun_fu4964807982371754893VEBT_o ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: set_real > fun_fu2614943698090592837real_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: set_nat > fun_fu7243318134412071974_nat_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: set_nat > fun_fu4988404976222497994_nat_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: set_nat > fun_fu2435797646934184285_nat_o ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: set_nat > fun_fu2828149587716756511BT_nat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_VE597728521808591204at_nat > fun_na6208021027755453088BT_nat ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: set_VEBT_VEBT > fun_fu4061198407372677825at_nat ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: set_nat > fun_fu5289434066392675523_nat_o ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: set_nat > fun_fu1580565752866428867_nat_o ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: set_complex > fun_fu3074144567237359299plex_o ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_nat > fun_fu8096999557949053699_nat_o ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: set_nat > fun_fu4176442768802536380er_nat ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_Co3312552843764780407at_nat > fun_na5492297256294505719er_nat ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: set_Code_integer > fun_fu6309186030181755779eger_o ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: set_Code_integer > fun_fu1095011443884336175at_nat ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: set_complex > fun_fu720419882121075617plex_o ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_complex > fun_fu8871020438547428127omplex ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_VE2006399705683203104omplex > fun_co3541073130412562976omplex ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: set_VEBT_VEBT > fun_fu8266604298679398363omplex ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: set_complex > fun_fu3043630359712370243plex_o ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: set_complex > fun_fu7433039812794084273omplex ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: set_real > fun_fu1853463631764337203omplex ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: set_complex > fun_fu5273811730930669507plex_o ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: set_complex > fun_fu6938252901338612493omplex ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: set_nat > fun_fu3780232916559739075_nat_o ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: set_nat > fun_fu3656911019502402539omplex ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: set_complex > fun_fu4149201653251398851plex_o ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: set_complex > fun_fu7724386853639975089omplex ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: set_int > fun_fu2620374121714387907_int_o ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: set_int > fun_fu260611630652282675omplex ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: set_complex > fun_fu7012200369710426499plex_o ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: set_complex > fun_fu403589845504736640omplex ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_Co8905303033605938227omplex > fun_co1305099417915878579omplex ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: set_Code_integer > fun_fu9120999634463783171eger_o ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: set_Code_integer > fun_fu243871637272208593omplex ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: set_Pr1773385645901665561uint32 > fun_ui3478071520421546738nt32_o ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: set_Pr1312406793078187350et_nat > fun_fu8419763957888848259_nat_o ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: set_Pr4040394263813967220et_nat > fun_fu2611741329380808165_nat_o ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_nat_nat > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_uint32 > fun_nat_uint32 ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_nat_uint32 > fun_nat_uint32 ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: set_VEBT_VEBT > fun_fu6775762051927458541VEBT_o ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: set_VEBT_VEBT > fun_fu7387521734868505405VEBT_o ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: set_real > fun_fu1352006120924326345real_o ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: set_real > fun_fu9205927727435512653real_o ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: set_nat > fun_fu8157887890162707073_nat_o ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: set_nat > fun_fu1829967339744774625_nat_o ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: set_int > fun_fu4320861794972987337_int_o ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: set_int > fun_fu6214582427498676685_int_o ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: set_complex > fun_fu5921220358231560645plex_o ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: set_complex > fun_fu5587198656263613639plex_o ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: set_Code_integer > fun_fu2470101743130883815eger_o ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: set_Code_integer > fun_fu2782498379308227194eger_o ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: set_VEBT_VEBT > fun_fu6610162456291744429VEBT_o ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: set_VEBT_VEBT > fun_fu8270842325377534013VEBT_o ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: set_VEBT_VEBT > fun_fu3470462164674526041VEBT_o ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: set_VEBT_VEBT > fun_fu7917602684971597757VEBT_o ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: set_complex > fun_fu2497916604042137945plex_o ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: set_complex > fun_fu6925816717872577647plex_o ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: set_Code_integer > fun_fu4523568072766896123eger_o ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: set_Code_integer > fun_fu7141499545569947298eger_o ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: set_VEBT_VEBT > fun_fu7910578496649732265VEBT_o ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: set_VEBT_VEBT > fun_fu7091630716096207293VEBT_o ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: set_real > fun_fu583245617705562381real_o ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: set_real > fun_fu6415922825302012245real_o ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: set_nat > fun_fu1116345184836134341_nat_o ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: set_nat > fun_fu3109912920687205353_nat_o ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: set_int > fun_fu2934462529731303693_int_o ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: set_int > fun_fu607692679349591509_int_o ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_list_nat_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_list_nat_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_set_nat_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_set_nat_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_nat_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_int_nat > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_complex_nat > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_Code_integer_nat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_Code_integer_nat > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_VEBT_VEBT_uint32 > fun_fu4239552856239149349uint32 ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_real_uint32 > fun_fu5616893895707965165uint32 ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_nat_uint32 > fun_fu5445312597681047533uint32 ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_int_uint32 > fun_fu4828966935367951085uint32 ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_complex_uint32 > fun_fu7155343958083200365uint32 ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: set_Code_integer > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_Co1554742298434871494uint32 > fun_fu7156106677247248941uint32 ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_VEBT_VEBT_real > fun_fu4911191927098811621T_real ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: vEBT_VEBT > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: vEBT_VEBT > fun_real_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: vEBT_VEBT > fun_int_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: nat > fun_VEBT_VEBT_nat ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: nat > fun_real_nat ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: int > fun_VEBT_VEBT_int ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_o_fun_nat_o ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: $o > fun_int_o ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: vEBT_VEBT > fun_o_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: set_VEBT_VEBT > fun_fu3657114993386715166VEBT_o ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: set_VEBT_VEBT > fun_fu6442816265145699251at_nat ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: set_real > fun_fu345129101432242798real_o ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: set_real > fun_fu2000997864302877705at_nat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: set_int > fun_fu2109032829546145134_int_o ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: set_int > fun_fu6978439502566113929at_nat ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: set_complex > fun_fu3884678914393021802plex_o ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: set_complex > fun_fu2213316472930114701at_nat ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: set_Code_integer > fun_fu5459709170960931212eger_o ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: set_Code_integer > fun_fu4917398172150038123at_nat ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: set_VEBT_VEBT > fun_fu5653810556089400794VEBT_o ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: set_VEBT_VEBT > fun_fu1645462851954096879omplex ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: set_real > fun_fu7786634604493040562real_o ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: set_real > fun_fu3142401395038748801omplex ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: set_nat > fun_fu2477617856857796458_nat_o ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: set_nat > fun_fu4641228418675179337omplex ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: set_int > fun_fu2764018104479922_int_o ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: set_int > fun_fu4036257109736896001omplex ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: set_Code_integer > fun_fu1571427466474281168eger_o ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: set_Code_integer > fun_fu1327305721226202083omplex ).

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

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: rat > fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_VEBT_VEBT_nat > fun_fu5238640373165125763BT_nat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: set_VEBT_VEBT > fun_fu5952108125419803843VEBT_o ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: set_VEBT_VEBT > fun_fu5952108125419803843VEBT_o ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: set_VEBT_VEBT > fun_fu6597202260156715244BT_nat ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: set_VEBT_VEBT > fun_fu2586499517409747105VEBT_o ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: set_real > fun_fu1979433296179983141real_o ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: set_real > fun_fu5265565954708106524BT_nat ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: set_real > fun_fu4123916144677202081real_o ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: set_VEBT_VEBT > fun_fu4730982365906966045VEBT_o ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: set_VEBT_VEBT > fun_fu6045111669986222662al_nat ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: set_real > fun_fu2513649851929744963real_o ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: set_real > fun_fu2513649851929744963real_o ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: set_real > fun_fu5295805721263303294al_nat ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: set_VEBT_VEBT > fun_fu7172300973527033475VEBT_o ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: set_literal > fun_fu7535628803236960835eral_o ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: set_literal > fun_fu1477312217196428428BT_nat ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: set_real > fun_fu1367744680140325601real_o ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: set_literal > fun_fu9030114850346191709eral_o ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: set_literal > fun_fu8777171035612306598al_nat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: set_literal > fun_fu2995320126720941571eral_o ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: set_VEBT_VEBT > fun_fu2631992297011014211VEBT_o ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: set_VEBT_VEBT > fun_fu6990504602265347692al_nat ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: set_literal > fun_fu3694550398032282849eral_o ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: set_real > fun_fu5255552264681192549real_o ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: set_real > fun_fu6805844602453103580al_nat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: set_literal > fun_fu2886836661332225987eral_o ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: set_literal > fun_fu2886836661332225987eral_o ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: set_literal > fun_fu5188895143799345676al_nat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: set_int > fun_fu1995833500178483365_int_o ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: set_int > fun_fu9044419119173484188BT_nat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: set_int > fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: set_complex > fun_se823489619503225858plex_o ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: set_Code_integer > fun_se5529282185252607396eger_o ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: set_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: set_VEBT_VEBT > fun_list_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: set_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: set_real > fun_list_real_o ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: set_o > fun_list_o_o ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: uint32 > fun_na7952645553354447727uint32 ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: uint32 > fun_na7952645553354447727uint32 ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_list_nat_nat > fun_li1025265955262268170_nat_o ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_set_nat_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_int_nat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_complex_nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_Code_integer_nat > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_VEBT_VEBT_nat > fun_fu8888284581464801555uint32 ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: set_VEBT_VEBT > fun_fu2605535626718998937uint32 ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_real_nat > fun_fu3635924368386386123uint32 ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: set_real > fun_fu628837400785242341uint32 ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_literal_nat > fun_fu7951284435484807859uint32 ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: set_literal > fun_fu5155170826040139294eral_o ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: set_literal > fun_fu1741543120565643833uint32 ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_VE5569070535936029429omplex > fun_fu3713327175455430069uint32 ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: set_VEBT_VEBT > fun_fu8675499575469611315uint32 ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_real_complex > fun_fu8010181880512525481uint32 ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: set_real > fun_fu8635684725810342015uint32 ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_literal_complex > fun_fu1840471026826697813uint32 ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: set_literal > fun_fu3916548891184232922eral_o ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: set_literal > fun_fu7431868672496406995uint32 ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_VE9219535225822561060nteger > fun_fu416210109593575622uint32 ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: set_VEBT_VEBT > fun_fu4063877677188701752VEBT_o ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: set_VEBT_VEBT > fun_fu3301057381668805568uint32 ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_re6978948838062957830nteger > fun_fu3468337015953762776uint32 ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: set_real > fun_fu1538760985949788884real_o ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: set_real > fun_fu6136972387607257868uint32 ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_li486744618827423812nteger > fun_fu5567281951138939366uint32 ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: set_literal > fun_fu3119719612471764280eral_o ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: set_literal > fun_fu128286754428327008uint32 ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_nat > fun_fu2798604719624125935uint32 ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: set_nat > fun_fu7243318134412071974_nat_o ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: set_nat > fun_fu923462241246005001uint32 ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_list_nat_nat > fun_fu4472246792138249213_nat_o ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_set_nat_nat > fun_fu1237770272497958833_nat_o ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_nat_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_int_nat > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_complex_nat > fun_fu6122667721776870753plex_o ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: fun_Code_integer_nat > fun_fu2182969643484777475eger_o ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: set_nat > fun_fu2604139981580899469ex_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: set_nat > fun_fu8277437853836697484_nat_o ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: set_nat > fun_fu2612476005019691883er_nat ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: set_nat > fun_fu7718009545821314414_nat_o ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: set_nat > fun_fu5979305884941020809nt_nat ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: set_complex > fun_fu3026973302150170441omplex ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: set_complex > fun_fu2219421879459728592plex_o ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: set_complex > fun_fu5234719464434624227omplex ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: set_complex > fun_fu7258100167511774898plex_o ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: set_complex > fun_fu7958774293462520321omplex ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: set_nat > fun_fu1144917263530448453x_real ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: set_nat > fun_fu622853494143631011r_real ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: set_nat > fun_fu28256929460833089t_real ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: set_int > fun_fu5178197999446659393at_int ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_int_uint32 > fun_int_o ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_complex_uint32 > fun_complex_o ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_Co1554742298434871494uint32 > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_Co3960896333048504454r_real > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_Code_integer_rat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_int_nat > fun_int_o ).

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

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_literal_nat > fun_literal_nat ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_list_nat_nat > fun_list_nat_nat ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_set_nat_nat > fun_set_nat_nat ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: set_nat > fun_fu7076988630286034496BT_nat ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_assn_fun_assn_o ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_assn_fun_assn_o ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_se5038631536132265326et_o_o ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_se5038631536132265326et_o_o ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: uint32 > fun_na5363892732107265798uint32 ).

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

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_complex_real > fun_re2354894731504631552x_real ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_Co3960896333048504454r_real > fun_re7921287988085964207r_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_complex_rat > fun_ra2209366104776213208ex_rat ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_Code_integer_rat > fun_ra3321464020636160263er_rat ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_complex_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_Code_integer_int > fun_in2718163036992669103er_int ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_int_real > fun_re7457141977439324286t_real ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: set_VEBT_VEBT > fun_list_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: set_real > fun_list_real_o ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: set_o > fun_list_o_o ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: set_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: set_complex > fun_list_complex_o ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: set_Code_integer > fun_li1325434182413301700eger_o ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: set_int > fun_list_int_o ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: vEBT_VEBT > fun_fu4141963935195952137T_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: real > fun_fu9132278742410140537l_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: code_integer > fun_fu8135725022974991116r_real ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: vEBT_VEBT > fun_fu4463953228700081077BT_rat ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: real > fun_fu7187499828406537085al_rat ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: code_integer > fun_fu8767785434844327952er_rat ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: vEBT_VEBT > fun_fu5586652545743564717BT_nat ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: real > fun_fu3768230810189712741al_nat ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: set_literal > fun_na6960352521224741070eral_o ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: set_list_nat > fun_na4138138649119996554_nat_o ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: set_set_nat > fun_na6948449159361463396_nat_o ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: set_int > fun_na7306406234920378942_int_o ).

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

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: vEBT_VEBT > fun_VE9201042017749490615o_assn ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: nat > fun_se33410294359761863t_assn ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: nat > fun_se33410294359761863t_assn ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: list_VEBT_VEBT > fun_ar1453094831611173103i_assn ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: nat > fun_se8971127763498933158T_assn ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: fun_nat_real ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_real_fun_int_o ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_real_o ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: nat > fun_na8563630686737515764_nat_o ).

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

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: nat > fun_se292914136812062355o_assn ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: nat > fun_se7717841832296886164t_assn ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: nat > fun_se7717841832296886164t_assn ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_real_o ).

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

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

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: nat > fun_li7327393198025160363T_VEBT ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: option4927543243414619207at_nat > fun_na3226088584868508064T_VEBT ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: nat > fun_li7327393198025160363T_VEBT ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: nat > fun_li9150060874205195607on_nat ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_real_real ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: fun_nat_real ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: fun_o_fun_nat_nat ).

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_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > fun_Pr6460944699146269935_nat_o ).

tff(sy_c_Assertions_Oentails,type,
    entails: ( assn * assn ) > $o ).

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

tff(sy_c_Assertions_Oex__assn_001t__VEBT____Definitions__OVEBT,type,
    ex_assn_VEBT_VEBT: fun_VEBT_VEBT_assn > assn ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: fun_o_assn ).

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

tff(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: fun_Pr6460944699146269935_nat_o > fun_fu8419763957888848259_nat_o ).

tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: fun_Pr3358449090565925198_nat_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re5089333283451836215nt_o_o: ( fun_int_fun_int_o * fun_o_fun_o_o ) > fun_fu6971708032831053102nt_o_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re4712519889275205905nt_int: ( fun_int_fun_int_o * fun_int_fun_int_o ) > fun_fu4208408993564413984_int_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re3715656647883201625at_nat: ( fun_int_fun_int_o * fun_nat_fun_nat_o * fun_int_nat * fun_int_nat ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_M_Eo_J_001t__fun_It__Int__Oint_M_Eo_J,type,
    bNF_re8654322613993568759_int_o: ( fun_int_fun_int_o * fun_fu6971708032831053102nt_o_o * fun_int_fun_int_o * fun_int_fun_int_o ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re5717476210804479843nt_int: ( fun_int_fun_int_o * fun_fu4208408993564413984_int_o * fun_int_fun_int_int * fun_int_fun_int_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    bNF_re8347998229424194295_nat_o: ( fun_int_fun_int_o * fun_fu7564235448449272366at_o_o * fun_int_fun_nat_o * fun_int_fun_nat_o ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
    bNF_re4705727531993890431at_o_o: ( fun_nat_fun_nat_o * fun_o_fun_o_o ) > fun_fu7564235448449272366at_o_o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re6650684261131312217nt_int: ( fun_nat_fun_nat_o * fun_int_fun_int_o * fun_nat_int * fun_nat_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > fun_fu9201817095025044880_nat_o ).

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tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__fun_It__Num__Onum_Mt__Int__Oint_J_001t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
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tff(sy_c_Binomial_Obinomial,type,
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tff(sy_c_Bit__Operations_Oconcat__bit,type,
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tff(sy_c_Bits__Integer_Ointeger__set__bit,type,
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tff(sy_c_Bits__Integer_Ointeger__shiftl,type,
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tff(sy_c_Bits__Integer_Ointeger__shiftr,type,
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tff(sy_c_Code__Numeral_Obit__cut__integer,type,
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tff(sy_c_Code__Numeral_Odivmod__abs,type,
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tff(sy_c_Code__Numeral_Odivmod__integer,type,
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tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
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tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
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tff(sy_c_Code__Numeral_Ointeger__of__num,type,
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tff(sy_c_Code__Numeral_Onat__of__integer,type,
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tff(sy_c_Code__Numeral_Onegative,type,
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tff(sy_c_Code__Numeral_Onum__of__integer,type,
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tff(sy_c_Code__Numeral_Opositive,type,
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tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
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tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
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tff(sy_c_Complex_Ocomplex_OComplex,type,
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tff(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
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tff(sy_c_Divides_Oadjust__mod,type,
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tff(sy_c_Divides_Odivmod__nat,type,
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tff(sy_c_Divides_Oeucl__rel__int,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
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tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
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tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
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tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
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tff(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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tff(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
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tff(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
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tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
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tff(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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    lattic4901227151466704046nteger: set_Code_integer > code_integer ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
    lattic8263393255366662781ax_int: set_int > int ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: fun_set_nat_nat ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Num__Onum,type,
    lattic4823215512031491691ax_num: set_num > num ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Rat__Orat,type,
    lattic7630753665789217321ax_rat: set_rat > rat ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Real__Oreal,type,
    lattic4275903605611617917x_real: set_real > real ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001_Eo,type,
    lattic1973801136483472281_Min_o: fun_set_o_o ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Code____Numeral__Ointeger,type,
    lattic1063845414844153500nteger: set_Code_integer > code_integer ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Int__Oint,type,
    lattic8718645017227715691in_int: set_int > int ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
    lattic8721135487736765967in_nat: fun_set_nat_nat ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Num__Onum,type,
    lattic5278467273892544601in_num: set_num > num ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Rat__Orat,type,
    lattic8086005427650270231in_rat: set_rat > rat ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Real__Oreal,type,
    lattic3629708407755379051n_real: set_real > real ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Rat__Orat,type,
    lattic2140725968369957399_o_rat: fun_o_rat > fun_set_o_o ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Real__Oreal,type,
    lattic8697145971487455083o_real: fun_o_real > fun_set_o_o ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Code____Numeral__Ointeger_001t__Rat__Orat,type,
    lattic5439806495466278992er_rat: ( fun_Code_integer_rat * set_Code_integer ) > code_integer ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Code____Numeral__Ointeger_001t__Real__Oreal,type,
    lattic2659822949269061924r_real: ( fun_Co3960896333048504454r_real * set_Code_integer ) > code_integer ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Num__Onum,type,
    lattic1922116423962787043ex_num: ( fun_complex_num * set_complex ) > complex ).

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

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    lattic8794016678065449205x_real: ( fun_complex_real * set_complex ) > complex ).

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__VEBT____Definitions__OVEBT_001t__Num__Onum,type,
    lattic3331990488459210229BT_num: ( fun_VEBT_VEBT_num * set_VEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
    lattic6139528642216935859BT_rat: ( fun_VEBT_VEBT_rat * set_VEBT_VEBT ) > vEBT_VEBT ).

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_001_Eo,type,
    append_o: ( list_o * list_o ) > list_o ).

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_Obutlast_001_Eo,type,
    butlast_o: list_o > list_o ).

tff(sy_c_List_Obutlast_001t__Int__Oint,type,
    butlast_int: list_int > list_int ).

tff(sy_c_List_Obutlast_001t__Nat__Onat,type,
    butlast_nat: list_nat > list_nat ).

tff(sy_c_List_Obutlast_001t__Real__Oreal,type,
    butlast_real: list_real > list_real ).

tff(sy_c_List_Obutlast_001t__VEBT____BuildupMemImp__OVEBTi,type,
    butlast_VEBT_VEBTi: list_VEBT_VEBTi > list_VEBT_VEBTi ).

tff(sy_c_List_Obutlast_001t__VEBT____Definitions__OVEBT,type,
    butlast_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).

tff(sy_c_List_Oconcat_001_Eo,type,
    concat_o: list_list_o > list_o ).

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

tff(sy_c_List_Oconcat_001t__Real__Oreal,type,
    concat_real: list_list_real > list_real ).

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

tff(sy_c_List_Ocount__list_001_Eo,type,
    count_list_o: list_o > fun_o_nat ).

tff(sy_c_List_Ocount__list_001t__Code____Numeral__Ointeger,type,
    count_3970941599679287265nteger: list_Code_integer > fun_Code_integer_nat ).

tff(sy_c_List_Ocount__list_001t__Complex__Ocomplex,type,
    count_list_complex: list_complex > fun_complex_nat ).

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

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

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

tff(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
    count_list_set_nat: list_set_nat > fun_set_nat_nat ).

tff(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
    count_list_VEBT_VEBT: list_VEBT_VEBT > fun_VEBT_VEBT_nat ).

tff(sy_c_List_Odistinct_001_Eo,type,
    distinct_o: fun_list_o_o ).

tff(sy_c_List_Odistinct_001t__Code____Numeral__Ointeger,type,
    distin1543349897113766820nteger: list_Code_integer > $o ).

tff(sy_c_List_Odistinct_001t__Complex__Ocomplex,type,
    distinct_complex: list_complex > $o ).

tff(sy_c_List_Odistinct_001t__Int__Oint,type,
    distinct_int: list_int > $o ).

tff(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: fun_list_nat_o ).

tff(sy_c_List_Odistinct_001t__Num__Onum,type,
    distinct_num: list_num > $o ).

tff(sy_c_List_Odistinct_001t__Rat__Orat,type,
    distinct_rat: list_rat > $o ).

tff(sy_c_List_Odistinct_001t__Real__Oreal,type,
    distinct_real: fun_list_real_o ).

tff(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
    distinct_set_nat: list_set_nat > $o ).

tff(sy_c_List_Odistinct_001t__VEBT____BuildupMemImp__OVEBTi,type,
    distinct_VEBT_VEBTi: list_VEBT_VEBTi > $o ).

tff(sy_c_List_Odistinct_001t__VEBT____Definitions__OVEBT,type,
    distinct_VEBT_VEBT: fun_list_VEBT_VEBT_o ).

tff(sy_c_List_Oenumerate_001_Eo,type,
    enumerate_o: ( nat * list_o ) > list_P7333126701944960589_nat_o ).

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__Real__Oreal,type,
    enumerate_real: ( nat * list_real ) > list_P3644420460460130531t_real ).

tff(sy_c_List_Oenumerate_001t__VEBT____BuildupMemImp__OVEBTi,type,
    enumerate_VEBT_VEBTi: ( nat * list_VEBT_VEBTi ) > list_P2320588648998582380_VEBTi ).

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

tff(sy_c_List_Ofilter_001t__Nat__Onat,type,
    filter_nat2: ( fun_nat_o * list_nat ) > list_nat ).

tff(sy_c_List_Ofoldl_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    foldl_assn_assn: ( fun_as3983628580203722482n_assn * assn * list_assn ) > assn ).

tff(sy_c_List_Ofoldl_001t__Int__Oint_001t__Int__Oint,type,
    foldl_int_int: ( fun_int_fun_int_int * int * list_int ) > int ).

tff(sy_c_List_Ofoldl_001t__Nat__Onat_001_Eo,type,
    foldl_nat_o: ( fun_nat_fun_o_nat * nat * list_o ) > nat ).

tff(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Nat__Onat,type,
    foldl_nat_nat: ( fun_nat_fun_nat_nat * nat ) > fun_list_nat_nat ).

tff(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Real__Oreal,type,
    foldl_nat_real: ( fun_nat_fun_real_nat * nat * list_real ) > nat ).

tff(sy_c_List_Ofoldl_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    foldl_nat_VEBT_VEBT: ( fun_na6208021027755453088BT_nat * nat * list_VEBT_VEBT ) > nat ).

tff(sy_c_List_Ofoldl_001t__Rat__Orat_001t__Rat__Orat,type,
    foldl_rat_rat: ( fun_rat_fun_rat_rat * rat * list_rat ) > rat ).

tff(sy_c_List_Ofoldl_001t__Real__Oreal_001t__Real__Oreal,type,
    foldl_real_real: ( fun_re7681423839302404990l_real * real * list_real ) > real ).

tff(sy_c_List_Ofoldl_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    foldl_uint32_uint32: ( fun_ui2637848667930195262uint32 * uint32 * list_uint32 ) > uint32 ).

tff(sy_c_List_Ofoldr_001_Eo_001t__Nat__Onat,type,
    foldr_o_nat: ( fun_o_fun_nat_nat * list_o ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
    foldr_nat_nat: ( fun_nat_fun_nat_nat * list_nat ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Nat__Onat,type,
    foldr_real_nat: ( fun_real_fun_nat_nat * list_real ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
    foldr_real_real: ( fun_re7681423839302404990l_real * list_real ) > fun_real_real ).

tff(sy_c_List_Ofoldr_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    foldr_VEBT_VEBT_nat: ( fun_VE597728521808591204at_nat * list_VEBT_VEBT ) > fun_nat_nat ).

tff(sy_c_List_Olast_001_Eo,type,
    last_o: fun_list_o_o ).

tff(sy_c_List_Olast_001t__Int__Oint,type,
    last_int: list_int > int ).

tff(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: fun_list_nat_nat ).

tff(sy_c_List_Olast_001t__Real__Oreal,type,
    last_real: list_real > real ).

tff(sy_c_List_Olast_001t__VEBT____BuildupMemImp__OVEBTi,type,
    last_VEBT_VEBTi: list_VEBT_VEBTi > vEBT_VEBTi ).

tff(sy_c_List_Olast_001t__VEBT____Definitions__OVEBT,type,
    last_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT ).

tff(sy_c_List_Olenlex_001_Eo,type,
    lenlex_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).

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

tff(sy_c_List_Olenlex_001t__Nat__Onat,type,
    lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

tff(sy_c_List_Olenlex_001t__Real__Oreal,type,
    lenlex_real: set_Pr6218003697084177305l_real > set_Pr611352295856513593t_real ).

tff(sy_c_List_Olenlex_001t__Uint32__Ouint32,type,
    lenlex_uint32: set_Pr1773385645901665561uint32 > set_Pr2258164808687509945uint32 ).

tff(sy_c_List_Olenlex_001t__VEBT____Definitions__OVEBT,type,
    lenlex_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

tff(sy_c_List_Olex_001_Eo,type,
    lex_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).

tff(sy_c_List_Olex_001t__Int__Oint,type,
    lex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

tff(sy_c_List_Olex_001t__Nat__Onat,type,
    lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

tff(sy_c_List_Olex_001t__Real__Oreal,type,
    lex_real: set_Pr6218003697084177305l_real > set_Pr611352295856513593t_real ).

tff(sy_c_List_Olex_001t__Uint32__Ouint32,type,
    lex_uint32: set_Pr1773385645901665561uint32 > set_Pr2258164808687509945uint32 ).

tff(sy_c_List_Olex_001t__VEBT____BuildupMemImp__OVEBTi,type,
    lex_VEBT_VEBTi: set_Pr2227491710730465451_VEBTi > set_Pr4797120415438012619_VEBTi ).

tff(sy_c_List_Olex_001t__VEBT____Definitions__OVEBT,type,
    lex_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001_Eo,type,
    linord3142498349692569832_set_o: set_o > list_o ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Code____Numeral__Ointeger,type,
    linord2324613341767563021nteger: set_Code_integer > list_Code_integer ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Int__Oint,type,
    linord2612477271533052124et_int: set_int > list_int ).

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

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Num__Onum,type,
    linord8395671565052656842et_num: set_num > list_num ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Rat__Orat,type,
    linord1979837681955606664et_rat: set_rat > list_rat ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Real__Oreal,type,
    linord4252657396651189596t_real: set_real > list_real ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__String__Oliteral,type,
    linord2913955441264437540iteral: set_literal > list_literal ).

tff(sy_c_List_Olist_OCons_001_Eo,type,
    cons_o: ( $o * list_o ) > list_o ).

tff(sy_c_List_Olist_OCons_001t__Code____Numeral__Ointeger,type,
    cons_Code_integer: ( code_integer * list_Code_integer ) > list_Code_integer ).

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_I_Eo_M_Eo_J,type,
    cons_P8766293264717362397od_o_o: ( product_prod_o_o * list_P4002435161011370285od_o_o ) > list_P4002435161011370285od_o_o ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    cons_P1455986808126089405_o_int: ( product_prod_o_int * list_P3795440434834930179_o_int ) > list_P3795440434834930179_o_int ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    cons_P5633837827635286113_o_nat: ( product_prod_o_nat * list_P6285523579766656935_o_nat ) > list_P6285523579766656935_o_nat ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
    cons_P7321330006258091179_int_o: ( product_prod_int_o * list_P5087981734274514673_int_o ) > list_P5087981734274514673_int_o ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    cons_P3334398858971670639nt_int: ( product_prod_int_int * list_P5707943133018811711nt_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    cons_P7512249878480867347nt_nat: ( product_prod_int_nat * list_P8198026277950538467nt_nat ) > list_P8198026277950538467nt_nat ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    cons_P9142372351690779143_nat_o: ( product_prod_nat_o * list_P7333126701944960589_nat_o ) > list_P7333126701944960589_nat_o ).

tff(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    cons_P2335045147070616083at_int: ( product_prod_nat_int * list_P3521021558325789923at_int ) > list_P3521021558325789923at_int ).

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__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    cons_P3149448846263281007uint32: ( produc827990862158126777uint32 * list_P3069071885182933823uint32 ) > list_P3069071885182933823uint32 ).

tff(sy_c_List_Olist_OCons_001t__Rat__Orat,type,
    cons_rat: ( rat * list_rat ) > list_rat ).

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__Uint32__Ouint32,type,
    cons_uint32: ( uint32 * list_uint32 ) > list_uint32 ).

tff(sy_c_List_Olist_OCons_001t__VEBT____BuildupMemImp__OVEBTi,type,
    cons_VEBT_VEBTi: ( vEBT_VEBTi * list_VEBT_VEBTi ) > list_VEBT_VEBTi ).

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_001_Eo,type,
    nil_o: list_o ).

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: fun_list_nat_nat ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001_Eo,type,
    map_nat_o: ( fun_nat_o * list_nat ) > list_o ).

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_Omap_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001_Eo,type,
    map_Pr7541730621154948341_o_o_o: ( fun_Pr6984036127010467572_o_o_o * list_P4002435161011370285od_o_o ) > list_o ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    map_VEBT_VEBT_nat: ( fun_VEBT_VEBT_nat * list_VEBT_VEBT ) > list_nat ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    map_VEBT_VEBT_real: ( fun_VEBT_VEBT_real * list_VEBT_VEBT ) > list_real ).

tff(sy_c_List_Olist_Oset_001_Eo,type,
    set_o2: fun_list_o_set_o ).

tff(sy_c_List_Olist_Oset_001t__Code____Numeral__Ointeger,type,
    set_Code_integer2: list_Code_integer > set_Code_integer ).

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

tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: fun_list_int_set_int ).

tff(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
    set_list_o2: list_list_o > set_list_o ).

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

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

tff(sy_c_List_Olist_Oset_001t__List__Olist_It__Real__Oreal_J,type,
    set_list_real2: list_list_real > set_list_real ).

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: fun_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_I_Eo_M_Eo_J,type,
    set_Product_prod_o_o2: list_P4002435161011370285od_o_o > set_Product_prod_o_o ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    set_Pr3864771053587467565et_nat: list_P9062070895058802706et_nat > set_Pr3948176798113811640et_nat ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
    set_Pr8694291782656941196_int_o: list_P5087981734274514673_int_o > set_Pr903927857289325719_int_o ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    set_Pr6647972299459129970nt_nat: list_P8198026277950538467nt_nat > set_Pr3448869479623346877nt_nat ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
    set_Pr112895574167722958t_real: list_P6863124054624500543t_real > set_Pr3538720872664544793t_real ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    set_Pr1470767568048878706at_int: list_P3521021558325789923at_int > set_Pr7995236796853374141at_int ).

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__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
    set_Pr8219819362198175822al_int: list_P4344331454722006975al_int > set_Pr1019928272762051225al_int ).

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

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

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

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    set_Pr2418681094996576974uint32: list_P3069071885182933823uint32 > set_Pr1773385645901665561uint32 ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    set_Pr7708085864119495200VEBT_o: list_P3126845725202233233VEBT_o > set_Pr3175402225741728619VEBT_o ).

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

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

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

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

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_Pr8131104793812927691et_nat: list_P8836981409019564976et_nat > set_Pr1312406793078187350et_nat ).

tff(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    set_Pr6610751079997246441et_nat: list_P4783563340545138382et_nat > set_Pr4040394263813967220et_nat ).

tff(sy_c_List_Olist_Oset_001t__Rat__Orat,type,
    set_rat2: list_rat > set_rat ).

tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: fun_li7683516459253347867t_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__String__Oliteral,type,
    set_literal2: list_literal > set_literal ).

tff(sy_c_List_Olist_Oset_001t__Uint32__Ouint32,type,
    set_uint322: list_uint32 > set_uint32 ).

tff(sy_c_List_Olist_Oset_001t__VEBT____BuildupMemImp__OVEBTi,type,
    set_VEBT_VEBTi2: list_VEBT_VEBTi > set_VEBT_VEBTi ).

tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: fun_li7721679215526719773T_VEBT ).

tff(sy_c_List_Olist_Oset_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    set_fu8796970219976790276_nat_o: list_f5222602229612168821_nat_o > set_fu3555266933746557007_nat_o ).

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

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_I_Eo_M_Eo_J,type,
    list_u1537252308907898773od_o_o: ( list_P4002435161011370285od_o_o * nat * product_prod_o_o ) > list_P4002435161011370285od_o_o ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    list_u9060326803697358356et_nat: ( list_P9062070895058802706et_nat * nat * produc3658429121746597890et_nat ) > list_P9062070895058802706et_nat ).

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

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

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    list_u2546161786664092711uint32: ( list_P3069071885182933823uint32 * nat * produc827990862158126777uint32 ) > list_P3069071885182933823uint32 ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    list_u30558089781959097_VEBTi: ( list_P785718909624839377_VEBTi * nat * produc3777764054643897931_VEBTi ) > list_P785718909624839377_VEBTi ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
    list_u9044509791230035014T_VEBT: ( list_P5988454224134618948T_VEBT * nat * produc2810682830582626868T_VEBT ) > list_P5988454224134618948T_VEBT ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    list_u636002643828906794_VEBTi: ( list_P735349106241217576_VEBTi * nat * produc3625547720036274456_VEBTi ) > list_P735349106241217576_VEBTi ).

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

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    list_u4153759941229589042et_nat: ( list_P8836981409019564976et_nat * nat * produc8799533983703829792et_nat ) > list_P8836981409019564976et_nat ).

tff(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    list_u3231058372064271824et_nat: ( list_P4783563340545138382et_nat * nat * produc3112576180687729086et_nat ) > list_P4783563340545138382et_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__Uint32__Ouint32,type,
    list_update_uint32: ( list_uint32 * nat * uint32 ) > list_uint32 ).

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

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

tff(sy_c_List_Olist__update_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    list_u2989257416974056029_nat_o: ( list_f5222602229612168821_nat_o * nat * fun_Pr6460944699146269935_nat_o ) > list_f5222602229612168821_nat_o ).

tff(sy_c_List_Olistrel1_001_Eo,type,
    listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).

tff(sy_c_List_Olistrel1_001t__Int__Oint,type,
    listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

tff(sy_c_List_Olistrel1_001t__Nat__Onat,type,
    listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

tff(sy_c_List_Olistrel1_001t__Real__Oreal,type,
    listrel1_real: set_Pr6218003697084177305l_real > set_Pr611352295856513593t_real ).

tff(sy_c_List_Olistrel1_001t__Uint32__Ouint32,type,
    listrel1_uint32: set_Pr1773385645901665561uint32 > set_Pr2258164808687509945uint32 ).

tff(sy_c_List_Olistrel1_001t__VEBT____BuildupMemImp__OVEBTi,type,
    listrel1_VEBT_VEBTi: set_Pr2227491710730465451_VEBTi > set_Pr4797120415438012619_VEBTi ).

tff(sy_c_List_Olistrel1_001t__VEBT____Definitions__OVEBT,type,
    listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

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

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__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__Real__Oreal_J,type,
    nth_Pr731366597535767232t_real: ( list_P6863124054624500543t_real * nat ) > produc679980390762269497t_real ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr5092243222550848201_VEBTi: ( list_P4447955379000644808_VEBTi * nat ) > produc2062302194595185346_VEBTi ).

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_M_Eo_J,type,
    nth_Pr112076138515278198_nat_o: ( list_P7333126701944960589_nat_o * nat ) > product_prod_nat_o ).

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__Real__Oreal_J,type,
    nth_Pr7767817059697521252t_real: ( list_P3644420460460130531t_real * nat ) > produc7716430852924023517t_real ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr3244165891152107629_VEBTi: ( list_P2320588648998582380_VEBTi * nat ) > produc214224863196444774_VEBTi ).

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__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    nth_Pr6139322253122866368uint32: ( list_P3069071885182933823uint32 * nat ) > produc827990862158126777uint32 ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_M_Eo_J,type,
    nth_Pr3306050735993963089EBTi_o: ( list_P8833571063612306856EBTi_o * nat ) > produc5014006835512566296EBTi_o ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Int__Oint_J,type,
    nth_Pr2733638074192486473Ti_int: ( list_P7392757774524453448Ti_int * nat ) > produc8927069083091599426Ti_int ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J,type,
    nth_Pr3433448822664029129i_real: ( list_P8536626330812492744i_real * nat ) > produc6680258955013199682i_real ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr6329974346453275474_VEBTi: ( list_P785718909624839377_VEBTi * nat ) > produc3777764054643897931_VEBTi ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr8725177398587324397T_VEBT: ( list_P5988454224134618948T_VEBT * nat ) > produc2810682830582626868T_VEBT ).

tff(sy_c_List_Onth_001t__Rat__Orat,type,
    nth_rat: list_rat > fun_nat_rat ).

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

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

tff(sy_c_List_Onth_001t__Uint32__Ouint32,type,
    nth_uint32: list_uint32 > fun_nat_uint32 ).

tff(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    nth_VEBT_VEBTi: list_VEBT_VEBTi > fun_nat_VEBT_VEBTi ).

tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_List_Oremove1_001_Eo,type,
    remove1_o: ( $o * list_o ) > list_o ).

tff(sy_c_List_Oremove1_001t__Code____Numeral__Ointeger,type,
    remove1_Code_integer: ( code_integer * list_Code_integer ) > list_Code_integer ).

tff(sy_c_List_Oremove1_001t__Int__Oint,type,
    remove1_int: ( int * list_int ) > list_int ).

tff(sy_c_List_Oremove1_001t__Nat__Onat,type,
    remove1_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Oremove1_001t__Num__Onum,type,
    remove1_num: ( num * list_num ) > list_num ).

tff(sy_c_List_Oremove1_001t__Rat__Orat,type,
    remove1_rat: ( rat * list_rat ) > list_rat ).

tff(sy_c_List_Oremove1_001t__Real__Oreal,type,
    remove1_real: ( real * list_real ) > list_real ).

tff(sy_c_List_Oremove1_001t__VEBT____Definitions__OVEBT,type,
    remove1_VEBT_VEBT: ( vEBT_VEBT * list_VEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_OremoveAll_001_Eo,type,
    removeAll_o: ( $o * list_o ) > list_o ).

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__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_001_Eo,type,
    rotate1_o: list_o > list_o ).

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

tff(sy_c_List_Orotate1_001t__Real__Oreal,type,
    rotate1_real: list_real > list_real ).

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_001_Eo,type,
    sorted_wrt_o: fun_o_fun_o_o > fun_list_o_o ).

tff(sy_c_List_Osorted__wrt_001t__Code____Numeral__Ointeger,type,
    sorted710888440204495920nteger: ( fun_Co2081214437782444048eger_o * list_Code_integer ) > $o ).

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

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

tff(sy_c_List_Osorted__wrt_001t__Num__Onum,type,
    sorted_wrt_num: ( fun_num_fun_num_o * list_num ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__Rat__Orat,type,
    sorted_wrt_rat: ( fun_rat_fun_rat_o * list_rat ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__Real__Oreal,type,
    sorted_wrt_real: fun_real_fun_real_o > fun_list_real_o ).

tff(sy_c_List_Osorted__wrt_001t__String__Oliteral,type,
    sorted_wrt_literal: ( fun_li4177005666941401710eral_o * list_literal ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    sorted9206477368072086664_VEBTi: ( fun_VE8636387394336989792EBTi_o * list_VEBT_VEBTi ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__VEBT____Definitions__OVEBT,type,
    sorted_wrt_VEBT_VEBT: fun_VE2962686792283380846VEBT_o > fun_list_VEBT_VEBT_o ).

tff(sy_c_List_Osubseqs_001_Eo,type,
    subseqs_o: list_o > list_list_o ).

tff(sy_c_List_Osubseqs_001t__Int__Oint,type,
    subseqs_int: list_int > list_list_int ).

tff(sy_c_List_Osubseqs_001t__Nat__Onat,type,
    subseqs_nat: list_nat > list_list_nat ).

tff(sy_c_List_Osubseqs_001t__Real__Oreal,type,
    subseqs_real: list_real > list_list_real ).

tff(sy_c_List_Osubseqs_001t__VEBT____Definitions__OVEBT,type,
    subseqs_VEBT_VEBT: list_VEBT_VEBT > list_list_VEBT_VEBT ).

tff(sy_c_List_Otake_001_Eo,type,
    take_o: ( nat * list_o ) > list_o ).

tff(sy_c_List_Otake_001t__Int__Oint,type,
    take_int: ( nat * list_int ) > list_int ).

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

tff(sy_c_List_Otake_001t__Num__Onum,type,
    take_num: ( nat * list_num ) > list_num ).

tff(sy_c_List_Otake_001t__Rat__Orat,type,
    take_rat: ( nat * list_rat ) > list_rat ).

tff(sy_c_List_Otake_001t__Real__Oreal,type,
    take_real: ( nat * list_real ) > list_real ).

tff(sy_c_List_Otake_001t__Set__Oset_It__Nat__Onat_J,type,
    take_set_nat: ( nat * list_set_nat ) > list_set_nat ).

tff(sy_c_List_Otake_001t__Uint32__Ouint32,type,
    take_uint32: ( nat * list_uint32 ) > list_uint32 ).

tff(sy_c_List_Otake_001t__VEBT____BuildupMemImp__OVEBTi,type,
    take_VEBT_VEBTi: ( nat * list_VEBT_VEBTi ) > list_VEBT_VEBTi ).

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

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

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

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

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_List_Ozip_001_Eo_001_Eo,type,
    zip_o_o: ( list_o * list_o ) > list_P4002435161011370285od_o_o ).

tff(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
    zip_o_int: ( list_o * list_int ) > list_P3795440434834930179_o_int ).

tff(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
    zip_o_nat: ( list_o * list_nat ) > list_P6285523579766656935_o_nat ).

tff(sy_c_List_Ozip_001_Eo_001t__Real__Oreal,type,
    zip_o_real: ( list_o * list_real ) > list_P5232166724548748803o_real ).

tff(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    zip_o_VEBT_VEBT: ( list_o * list_VEBT_VEBT ) > list_P7495141550334521929T_VEBT ).

tff(sy_c_List_Ozip_001t__Int__Oint_001_Eo,type,
    zip_int_o: ( list_int * list_o ) > list_P5087981734274514673_int_o ).

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

tff(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
    zip_int_VEBT_VEBTi: ( list_int * list_VEBT_VEBTi ) > list_P4447955379000644808_VEBTi ).

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__Nat__Onat_001_Eo,type,
    zip_nat_o: ( list_nat * list_o ) > list_P7333126701944960589_nat_o ).

tff(sy_c_List_Ozip_001t__Nat__Onat_001t__Int__Oint,type,
    zip_nat_int: ( list_nat * list_int ) > list_P3521021558325789923at_int ).

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

tff(sy_c_List_Ozip_001t__Real__Oreal_001_Eo,type,
    zip_real_o: ( list_real * list_o ) > list_P3595434254542482545real_o ).

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

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

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

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

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

tff(sy_c_List_Ozip_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    zip_uint32_uint32: ( list_uint32 * list_uint32 ) > list_P3069071885182933823uint32 ).

tff(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    zip_VEBT_VEBTi_o: ( list_VEBT_VEBTi * list_o ) > list_P8833571063612306856EBTi_o ).

tff(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    zip_VEBT_VEBTi_int: ( list_VEBT_VEBTi * list_int ) > list_P7392757774524453448Ti_int ).

tff(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    zip_VEBT_VEBTi_real: ( list_VEBT_VEBTi * list_real ) > list_P8536626330812492744i_real ).

tff(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    zip_VE793581609497812771_VEBTi: ( list_VEBT_VEBTi * list_VEBT_VEBTi ) > list_P785718909624839377_VEBTi ).

tff(sy_c_List_Ozip_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    zip_VE7413257051550508102T_VEBT: ( list_VEBT_VEBTi * list_VEBT_VEBT ) > list_P5988454224134618948T_VEBT ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    zip_VEBT_VEBT_o: ( list_VEBT_VEBT * list_o ) > list_P3126845725202233233VEBT_o ).

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

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

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

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    zip_VE6444338338598820466_VEBTi: ( list_VEBT_VEBT * list_VEBT_VEBTi ) > list_P735349106241217576_VEBTi ).

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_List_Ozip_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    zip_fu1498620252286093946et_nat: ( list_f5222602229612168821_nat_o * list_P9062070895058802706et_nat ) > list_P8836981409019564976et_nat ).

tff(sy_c_List_Ozip_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    zip_fu1756369540273021848et_nat: ( list_f5222602229612168821_nat_o * list_P8836981409019564976et_nat ) > list_P4783563340545138382et_nat ).

tff(sy_c_Misc_Obijective_001t__Int__Oint_001t__Int__Oint,type,
    bijective_int_int: set_Pr958786334691620121nt_int > $o ).

tff(sy_c_Misc_Obijective_001t__Nat__Onat_001t__Nat__Onat,type,
    bijective_nat_nat: set_Pr1261947904930325089at_nat > $o ).

tff(sy_c_Misc_Obijective_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    biject2923219584331114343uint32: set_Pr1773385645901665561uint32 > $o ).

tff(sy_c_Misc_Obijective_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    biject6574446902887564708et_nat: set_Pr1312406793078187350et_nat > $o ).

tff(sy_c_Misc_Obijective_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    biject4066639966817638850et_nat: set_Pr4040394263813967220et_nat > $o ).

tff(sy_c_Misc_Odflt__None__set_001_Eo,type,
    dflt_None_set_o: set_o > option_set_o ).

tff(sy_c_Misc_Odflt__None__set_001t__Int__Oint,type,
    dflt_None_set_int: set_int > option_set_int ).

tff(sy_c_Misc_Odflt__None__set_001t__Nat__Onat,type,
    dflt_None_set_nat: set_nat > option_set_nat ).

tff(sy_c_Misc_Omergesort__remdups_001_Eo,type,
    mergesort_remdups_o: list_o > list_o ).

tff(sy_c_Misc_Omergesort__remdups_001t__Int__Oint,type,
    merges1654122896337280797ps_int: list_int > list_int ).

tff(sy_c_Misc_Omergesort__remdups_001t__Nat__Onat,type,
    merges1656613366846331073ps_nat: list_nat > list_nat ).

tff(sy_c_Misc_Omergesort__remdups_001t__Num__Onum,type,
    merges7437317189856885515ps_num: list_num > list_num ).

tff(sy_c_Misc_Omergesort__remdups_001t__Rat__Orat,type,
    merges1021483306759835337ps_rat: list_rat > list_rat ).

tff(sy_c_Misc_Omergesort__remdups_001t__Real__Oreal,type,
    merges7559785487730971421s_real: list_real > list_real ).

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

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

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

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

tff(sy_c_Misc_Oslice_001t__VEBT____BuildupMemImp__OVEBTi,type,
    slice_VEBT_VEBTi: ( nat * nat * list_VEBT_VEBTi ) > list_VEBT_VEBTi ).

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

tff(sy_c_Misc_Othe__default_001t__Nat__Onat,type,
    the_default_nat: ( nat * option_nat ) > nat ).

tff(sy_c_Misc_Othe__default_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_de3812365490807259288at_nat: ( product_prod_nat_nat * option4927543243414619207at_nat ) > product_prod_nat_nat ).

tff(sy_c_Misc_Othe__default_001t__Set__Oset_I_Eo_J,type,
    the_default_set_o: ( set_o * option_set_o ) > set_o ).

tff(sy_c_Misc_Othe__default_001t__Set__Oset_It__Int__Oint_J,type,
    the_default_set_int: ( set_int * option_set_int ) > set_int ).

tff(sy_c_Misc_Othe__default_001t__Set__Oset_It__Nat__Onat_J,type,
    the_default_set_nat: ( set_nat * option_set_nat ) > set_nat ).

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_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Uint32__Ouint32,type,
    semiri2565882477558803405uint32: fun_nat_uint32 ).

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

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

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
    size_s2710708370519433104list_o: list_list_o > nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: fun_list_nat_nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Rat__Orat_J,type,
    size_size_list_rat: list_rat > nat ).

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__String__Oliteral_J,type,
    size_s2501651207091587910iteral: list_literal > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Uint32__Ouint32_J,type,
    size_s4844771616002835472uint32: list_uint32 > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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tff(sy_c_Nat__Bijection_Oset__decode,type,
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tff(sy_c_Nat__Bijection_Oset__encode,type,
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tff(sy_c_Nat__Bijection_Otriangle,type,
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tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Uint32__Ouint32,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Uint32__Ouint32,type,
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tff(sy_c_Num_Onum_OBit0,type,
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tff(sy_c_Num_Onum_OBit1,type,
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tff(sy_c_Num_Onum_OOne,type,
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tff(sy_c_Num_Onum_Osize__num,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Uint32__Ouint32,type,
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tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
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tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
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tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Uint32__Ouint32_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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tff(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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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__Option__Ooption_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__String__Oliteral,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Uint32__Ouint32,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_I_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Assertions__Oassn,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
    ord_max_Code_integer: code_integer > fun_Co8499238673658658103nteger ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: num > fun_num_num ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: real > fun_real_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_I_Eo_J,type,
    ord_max_set_o: ( set_o * set_o ) > set_o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: ( set_int * set_int ) > set_int ).

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Code____Numeral__Ointeger,type,
    ord_min_Code_integer: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
    ord_min_int: int > fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Orderings_Oordering__top_001t__Assertions__Oassn,type,
    ordering_top_assn: ( fun_assn_fun_assn_o * fun_assn_fun_assn_o * assn ) > $o ).

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

tff(sy_c_Orderings_Oordering__top_001t__Set__Oset_I_Eo_J,type,
    ordering_top_set_o: ( fun_se5038631536132265326et_o_o * fun_se5038631536132265326et_o_o ) > fun_set_o_o ).

tff(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Int__Oint_J,type,
    ordering_top_set_int: ( fun_se2051630309975736070_int_o * fun_se2051630309975736070_int_o ) > fun_set_int_o ).

tff(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
    ordering_top_set_nat: ( fun_se2170385732985038782_nat_o * fun_se2170385732985038782_nat_o ) > fun_set_nat_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Numeral____Type__Onum0_J,type,
    top_to3689904424835650196l_num0: set_Numeral_num0 ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
    top_top_set_literal: set_literal ).

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

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

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

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

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Power_Opower__class_Opower_001t__Uint32__Ouint32,type,
    power_power_uint32: uint32 > fun_nat_uint32 ).

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

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

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

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7507926704131184380et_nat: ( heap_e7401611519738050253t_unit * set_nat ) > produc3658429121746597890et_nat ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001_Eo,type,
    product_Pair_int_o: ( int * $o ) > product_prod_int_o ).

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__Real__Oreal,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
    produc8435520187683070743list_o: ( list_o * list_o ) > produc7102631898165422375list_o ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
    produc364263696895485585st_int: ( list_int * list_int ) > produc1186641810826059865st_int ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: ( list_nat * list_nat ) > produc1828647624359046049st_nat ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Real__Oreal_J_001t__List__Olist_It__Real__Oreal_J,type,
    produc1408950526243324945t_real: ( list_real * list_real ) > produc478978216448986841t_real ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__Uint32__Ouint32_J_001t__List__Olist_It__Uint32__Ouint32_J,type,
    produc7487160679990061969uint32: ( list_uint32 * list_uint32 ) > produc823665806539077721uint32 ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    produc4384243565435462691_VEBTi: ( list_VEBT_VEBTi * list_VEBT_VEBTi ) > produc3089554586268799851_VEBTi ).

tff(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    produc3897820843166775703T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > produc9211091688327510695T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
    product_Pair_nat_o: ( nat * $o ) > product_prod_nat_o ).

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__Real__Oreal,type,
    produc7837566107596912789t_real: ( nat * real ) > produc7716430852924023517t_real ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc3646306378393792727nt_int: ( product_prod_int_int * product_prod_int_int ) > produc1219242969750017639nt_int ).

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__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    produc6884863695460953815uint32: ( produc827990862158126777uint32 * produc827990862158126777uint32 ) > produc8822557026176459367uint32 ).

tff(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc8050396023069295193et_nat: ( produc8799533983703829792et_nat * produc8799533983703829792et_nat ) > produc8893060842726229537et_nat ).

tff(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    produc7768440062179809557et_nat: ( produc3112576180687729086et_nat * produc3112576180687729086et_nat ) > produc505541996239113949et_nat ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
    produc3179012173361985393al_int: ( real * int ) > produc8786904178792722361al_int ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: ( real * real ) > produc2422161461964618553l_real ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc6931449550656315951T_VEBT: ( real * vEBT_VEBT ) > produc3757001726724277373T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: ( set_nat * set_nat ) > produc7819656566062154093et_nat ).

tff(sy_c_Product__Type_OPair_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    produc1400373151660368625uint32: ( uint32 * uint32 ) > produc827990862158126777uint32 ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    produc8194178580519725514EBTi_o: ( vEBT_VEBTi * $o ) > produc5014006835512566296EBTi_o ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    produc7190175284220460154Ti_int: ( vEBT_VEBTi * int ) > produc8927069083091599426Ti_int ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    produc8457151488442208762i_real: ( vEBT_VEBTi * real ) > produc6680258955013199682i_real ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc436343169921013763_VEBTi: ( vEBT_VEBTi * vEBT_VEBTi ) > produc3777764054643897931_VEBTi ).

tff(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    produc7053807326796202854T_VEBT: ( vEBT_VEBTi * vEBT_VEBT ) > produc2810682830582626868T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc8721562602347293563VEBT_o: ( vEBT_VEBT * $o ) > produc334124729049499915VEBT_o ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8117437818029410057T_real: ( vEBT_VEBT * real ) > produc5170161368751668367T_real ).

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

tff(sy_c_Product__Type_OPair_001t__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__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc6017135503889078682et_nat: ( fun_Pr6460944699146269935_nat_o * produc3658429121746597890et_nat ) > produc8799533983703829792et_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
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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_001_Eo_001_Eo_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_Transitive__Closure_Otrancl_001t__Real__Oreal,type,
    transi1789104906590519371l_real: set_Pr6218003697084177305l_real > set_Pr6218003697084177305l_real ).

tff(sy_c_Transitive__Closure_Otrancl_001t__Uint32__Ouint32,type,
    transi3114468042090999947uint32: set_Pr1773385645901665561uint32 > set_Pr1773385645901665561uint32 ).

tff(sy_c_Transitive__Closure_Otrancl_001t__VEBT____BuildupMemImp__OVEBTi,type,
    transi2803566869205510612_VEBTi: set_Pr2227491710730465451_VEBTi > set_Pr2227491710730465451_VEBTi ).

tff(sy_c_Transitive__Closure_Otrancl_001t__VEBT____Definitions__OVEBT,type,
    transi8906537157094044885T_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr6192946355708809607T_VEBT ).

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

tff(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    type_d8366093980585677751er_int: ( fun_Code_integer_int * fun_int_Code_integer ) > fun_set_int_o ).

tff(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( fun_nat_o * nat * uint32 ) > uint32 ).

tff(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: fun_na3857091055356329634uint32 ).

tff(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint32: uint32 > word_N3645301735248828278l_num1 ).

tff(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: ( uint32 * uint32 ) > produc827990862158126777uint32 ).

tff(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: ( uint32 * code_integer ) > fun_o_uint32 ).

tff(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: uint32 > fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: uint32 > fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: uint32 > fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: uint32 > fun_Code_integer_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun_nat_fun_nat_o ).

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

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

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap_o ).

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

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap_o ).

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

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

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun_VE905575822219544865i_assn ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: fun_Pr3720955412185915546EBTi_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: ( vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: fun_VE8636387394336989792EBTi_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: ( vEBT_VEBTi * nat ) > heap_Time_Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: fun_VE8636387394336989792EBTi_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT ) > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: nat > fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: nat > fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: fun_VE6480751511521678896_nat_o ).

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
    vEBT_V1365221501068881998eletei: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
    vEBT_vebt_deletei: ( vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: fun_VE597728521808591204at_nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
    vEBT_VEBT_insert: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
    vEBT_Intf_vebt_assn: ( nat * set_nat ) > fun_VEBT_VEBTi_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6286945158656146733_VEBTi: ( set_nat * fun_o_9029690514724572677i_assn * list_o * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1319876754960170684T_VEBT: ( set_nat * fun_o_4353168661238401050T_assn * list_o * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001_Eo,type,
    vEBT_L3563379889750563018_int_o: ( set_nat * fun_int_fun_o_assn * list_int * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__Int__Oint,type,
    vEBT_L8888932350013902202nt_int: ( set_nat * fun_int_fun_int_assn * list_int * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__Nat__Onat,type,
    vEBT_L8891422820522952478nt_nat: ( set_nat * fun_int_fun_nat_assn * list_int * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L7077748017936769786t_real: ( set_nat * fun_in6446958447386763770l_assn * list_int * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L114188773329725699_VEBTi: ( set_nat * fun_in1903134756389476099i_assn * list_int * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2018189785592951398T_VEBT: ( set_nat * fun_in7755481285845205148T_assn * list_int * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7489483478785760935_VEBTi: ( set_nat * fun_na6467670216208870567i_assn * list_nat * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8511957252848910786T_VEBT: ( set_nat * fun_na4489259242115493880T_assn * list_nat * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7851252805511451907_VEBTi: ( set_nat * fun_re560840401387337987i_assn * list_real * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3095048238742455910T_VEBT: ( set_nat * fun_re8050108611700907676T_assn * list_real * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    vEBT_L3328983362619735041EBTi_o: ( set_nat * fun_VE9201042017749490615o_assn * list_VEBT_VEBTi * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    vEBT_L2806540629473551875Ti_int: ( set_nat * fun_VE5641251438918192003t_assn * list_VEBT_VEBTi * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    vEBT_L2809031099982602151Ti_nat: ( set_nat * fun_VE1225457145571040039t_assn * list_VEBT_VEBTi * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    vEBT_L7728200936804140803i_real: ( set_nat * fun_VE1885522831824361987l_assn * list_VEBT_VEBTi * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L886525131989349516_VEBTi: ( set_nat * fun_VE2561185598974572172i_assn * list_VEBT_VEBTi * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2497118539674116125T_VEBT: ( set_nat * fun_VE5577048379444288083T_assn * list_VEBT_VEBTi * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8650695023172932196BT_nat: ( set_nat * fun_VE4931415397359990460t_assn * list_VEBT_VEBT * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: ( set_nat * fun_VE905575822219544865i_assn * list_VEBT_VEBT * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3204528365124325536T_VEBT: ( set_nat * fun_VE1571388752190420990T_assn * list_VEBT_VEBT * list_VEBT_VEBT ) > assn ).

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

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1750719106661372127T_VEBT: ( fun_o_4353168661238401050T_assn * list_o * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Int__Oint,type,
    vEBT_L74593716426352029nt_int: ( fun_int_fun_int_assn * list_int * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Nat__Onat,type,
    vEBT_L77084186935402305nt_nat: ( fun_int_fun_nat_assn * list_int * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6235239671944049190_VEBTi: ( fun_in1903134756389476099i_assn * list_int * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1664421287176695555T_VEBT: ( fun_in7755481285845205148T_assn * list_int * list_VEBT_VEBT ) > assn ).

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

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8158188754432654943T_VEBT: ( fun_na4489259242115493880T_assn * list_nat * list_VEBT_VEBT ) > assn ).

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

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

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L4595930785310033027T_VEBT: ( fun_re8050108611700907676T_assn * list_real * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    vEBT_L8927591528087875366Ti_int: ( fun_VE5641251438918192003t_assn * list_VEBT_VEBTi * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    vEBT_L8930081998596925642Ti_nat: ( fun_VE1225457145571040039t_assn * list_VEBT_VEBTi * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1891944875198410415_VEBTi: ( fun_VE2561185598974572172i_assn * list_VEBT_VEBTi * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L7265847600308530106T_VEBT: ( fun_VE5577048379444288083T_assn * list_VEBT_VEBTi * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7489408758114837031VEBT_o: ( fun_VE2104784059010244834o_assn * list_VEBT_VEBT * list_o ) > assn ).

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

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

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

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

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat ) > fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option_nat * option_nat ) > $o ).

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

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

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

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

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun_VEBT_VEBT_real ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: ( vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: ( vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

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

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

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Uniqueness_OVEBT__internal_OperInsTrans,type,
    vEBT_V6289311342943941716sTrans: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: fun_Pr9018117161326028078_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: fun_Pr2028404973800837934_nat_o > fun_Pr4796943543331071514_nat_o ).

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

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    accp_P7675410724331315407_VEBTi: ( fun_Pr3720955412185915546EBTi_o * produc3625547720036274456_VEBTi ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    accp_P1394777952347926645et_nat: fun_Pr3358449090565925198_nat_o > fun_Pr1227177680593602867_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____BuildupMemImp__OVEBTi,type,
    accp_VEBT_VEBTi: fun_VE8636387394336989792EBTi_o > fun_VEBT_VEBTi_o ).

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

tff(sy_c_Word_Osigned__drop__bit_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    signed5000768011106662067l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

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

tff(sy_c_aa_001_Eo_001t__Assertions__Oassn,type,
    aa_o_assn: ( fun_o_assn * $o ) > assn ).

tff(sy_c_aa_001_Eo_001t__Code____Numeral__Ointeger,type,
    aa_o_Code_integer: ( fun_o_Code_integer * $o ) > code_integer ).

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

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

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

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

tff(sy_c_aa_001_Eo_001t__Uint32__Ouint32,type,
    aa_o_uint32: ( fun_o_uint32 * $o ) > uint32 ).

tff(sy_c_aa_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    aa_o_VEBT_VEBT: ( fun_o_VEBT_VEBT * $o ) > vEBT_VEBT ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_M_Eo_J,type,
    aa_o_fun_o_o: ( fun_o_fun_o_o * $o ) > fun_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_o_fun_nat_o: ( fun_o_fun_nat_o * $o ) > fun_nat_o ).

tff(sy_c_aa_001_Eo_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_o_fun_nat_nat: ( fun_o_fun_nat_nat * $o ) > fun_nat_nat ).

tff(sy_c_aa_001_Eo_001t__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J,type,
    aa_o_f7290096833689243702i_assn: ( fun_o_9029690514724572677i_assn * $o ) > fun_VEBT_VEBTi_assn ).

tff(sy_c_aa_001_Eo_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J,type,
    aa_o_f2789175432390837779T_assn: ( fun_o_4353168661238401050T_assn * $o ) > fun_VEBT_VEBT_assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001_Eo,type,
    aa_assn_o: ( fun_assn_o * assn ) > $o ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    aa_assn_assn: ( fun_assn_assn * assn ) > assn ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__Assertions__Oassn_M_Eo_J,type,
    aa_assn_fun_assn_o: ( fun_assn_fun_assn_o * assn ) > fun_assn_o ).

tff(sy_c_aa_001t__Assertions__Oassn_001t__fun_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    aa_ass954072293579886259n_assn: ( fun_as3983628580203722482n_assn * assn ) > fun_assn_assn ).

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

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Complex__Ocomplex,type,
    aa_Cod5621350732091595593omplex: ( fun_Co1900716531650368264omplex * code_integer ) > complex ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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tff(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    member568628332442017744T_VEBT: ( produc8243902056947475879T_VEBT * set_Pr6192946355708809607T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    member4513763875858039863et_nat: ( produc8799533983703829792et_nat * set_Pr1312406793078187350et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member9174161442226468181et_nat: ( produc3112576180687729086et_nat * set_Pr4040394263813967220et_nat ) > $o ).

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

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

tff(sy_c_member_001t__Set__Oset_I_Eo_J,type,
    member_set_o: ( set_o * set_set_o ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: ( set_int * set_set_int ) > $o ).

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

tff(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
    member_set_real: ( set_real * set_set_real ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    member_set_set_nat: ( set_set_nat * set_set_set_nat ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    member_set_VEBT_VEBT: ( set_VEBT_VEBT * set_set_VEBT_VEBT ) > $o ).

tff(sy_c_member_001t__String__Oliteral,type,
    member_literal: ( literal * set_literal ) > $o ).

tff(sy_c_member_001t__Uint32__Ouint32,type,
    member_uint32: ( uint32 * set_uint32 ) > $o ).

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

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

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    member7544390268018397336_nat_o: ( fun_Pr6460944699146269935_nat_o * set_fu3555266933746557007_nat_o ) > $o ).

tff(sy_v_a,type,
    a: heap_e7401611519738050253t_unit ).

tff(sy_v_b,type,
    b: set_nat ).

tff(sy_v_n,type,
    n: nat ).

tff(sy_v_r,type,
    r: vEBT_VEBTi ).

tff(sy_v_s,type,
    s: set_nat ).

tff(sy_v_t,type,
    t: vEBT_VEBT ).

tff(sy_v_x,type,
    x: nat ).

tff(sy_v_xa,type,
    xa: nat ).

% Relevant facts (9783)
tff(fact_0_invar__vebt__delete,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa)),Na) ) ).

% invar_vebt_delete
tff(fact_1_set__vebt__equal,axiom,
    ! [T_1: vEBT_VEBT,Na: nat,T_2: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T_1),Na)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T_2),Na)
       => ( ( T_1 = T_2 )
        <=> ( vEBT_set_vebt(T_1) = vEBT_set_vebt(T_2) ) ) ) ) ).

% set_vebt_equal
tff(fact_2_unique__tree,axiom,
    ! [T: vEBT_VEBT,Na: nat,Sa: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Sa),Na)
       => ( ( vEBT_set_vebt(T) = vEBT_set_vebt(Sa) )
         => ( Sa = T ) ) ) ) ).

% unique_tree
tff(fact_3_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_VEBT_insert(T),Xa)),Na) ) ).

% insert'_pres_valid
tff(fact_4_perIT__concat,axiom,
    ! [Sa: vEBT_VEBT,T: vEBT_VEBT,U: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,Sa),T)
     => ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,T),U)
       => aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,Sa),U) ) ) ).

% perIT_concat
tff(fact_5_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_6_uniquetree,axiom,
    ! [T: vEBT_VEBT,Na: nat,Sa: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Sa),Na)
       => ( ( vEBT_VEBT_set_vebt(T) = vEBT_VEBT_set_vebt(Sa) )
         => ( Sa = T ) ) ) ) ).

% uniquetree
tff(fact_7_Rep__assn__inject,axiom,
    ! [Xa: assn,Y: assn] :
      ( ( rep_assn(Xa) = rep_assn(Y) )
    <=> ( Xa = Y ) ) ).

% Rep_assn_inject
tff(fact_8_prod_Oinject,axiom,
    ! [X1: uint32,X2: uint32,Y1: uint32,Y2: uint32] :
      ( ( produc1400373151660368625uint32(X1,X2) = produc1400373151660368625uint32(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_9_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_10_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( product_Pair_int_int(X1,X2) = product_Pair_int_int(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_11_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X2: produc3658429121746597890et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y2: produc3658429121746597890et_nat] :
      ( ( produc6017135503889078682et_nat(X1,X2) = produc6017135503889078682et_nat(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_12_prod_Oinject,axiom,
    ! [X1: fun_Pr6460944699146269935_nat_o,X2: produc8799533983703829792et_nat,Y1: fun_Pr6460944699146269935_nat_o,Y2: produc8799533983703829792et_nat] :
      ( ( produc7291855603632349880et_nat(X1,X2) = produc7291855603632349880et_nat(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_13_old_Oprod_Oinject,axiom,
    ! [A: uint32,B: uint32,A2: uint32,B2: uint32] :
      ( ( produc1400373151660368625uint32(A,B) = produc1400373151660368625uint32(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_14_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_15_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( product_Pair_int_int(A,B) = product_Pair_int_int(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_16_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
      ( ( produc6017135503889078682et_nat(A,B) = produc6017135503889078682et_nat(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_17_old_Oprod_Oinject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
      ( ( produc7291855603632349880et_nat(A,B) = produc7291855603632349880et_nat(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_18_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,T),D)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),D) ) ).

% valid_eq
tff(fact_19_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,T),D) ) ).

% valid_eq1
tff(fact_20_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,T),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),D) ) ).

% valid_eq2
tff(fact_21_set__vebt__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xa)
      <=> aa_set_nat_o(member_nat(Xa),vEBT_set_vebt(T)) ) ) ).

% set_vebt_member
tff(fact_22_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa)),Y)
      <=> ( ( Xa != Y )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_23_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa)),Y)
      <=> ( ( Xa != Y )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_24_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xa)
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xa) ) ) ).

% valid_member_both_member_options
tff(fact_25_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xa)
      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xa) ) ) ).

% both_member_options_equiv_member
tff(fact_26_times__assn__raw_Ocases,axiom,
    ! [Xa: produc3112576180687729086et_nat] :
      ~ ! [P: fun_Pr6460944699146269935_nat_o,Q: fun_Pr6460944699146269935_nat_o,H: heap_e7401611519738050253t_unit,As: set_nat] : ( Xa != produc7291855603632349880et_nat(P,produc6017135503889078682et_nat(Q,produc7507926704131184380et_nat(H,As))) ) ).

% times_assn_raw.cases
tff(fact_27_prod__induct4,axiom,
    ! [P2: fun_Pr1227177680593602867_nat_o,Xa: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C: heap_e7401611519738050253t_unit,D2: set_nat] : aa_Pro2671238023309943196_nat_o(P2,produc7291855603632349880et_nat(A3,produc6017135503889078682et_nat(B3,produc7507926704131184380et_nat(C,D2))))
     => aa_Pro2671238023309943196_nat_o(P2,Xa) ) ).

% prod_induct4
tff(fact_28_prod__induct3,axiom,
    ! [P2: fun_Pr2620521676128474833_nat_o,Xa: produc8799533983703829792et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: heap_e7401611519738050253t_unit,C: set_nat] : aa_Pro5643009357358181178_nat_o(P2,produc6017135503889078682et_nat(A3,produc7507926704131184380et_nat(B3,C)))
     => aa_Pro5643009357358181178_nat_o(P2,Xa) ) ).

% prod_induct3
tff(fact_29_prod__induct3,axiom,
    ! [P2: fun_Pr1227177680593602867_nat_o,Xa: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C: produc3658429121746597890et_nat] : aa_Pro2671238023309943196_nat_o(P2,produc7291855603632349880et_nat(A3,produc6017135503889078682et_nat(B3,C)))
     => aa_Pro2671238023309943196_nat_o(P2,Xa) ) ).

% prod_induct3
tff(fact_30_prod__cases4,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C: heap_e7401611519738050253t_unit,D2: set_nat] : ( Y != produc7291855603632349880et_nat(A3,produc6017135503889078682et_nat(B3,produc7507926704131184380et_nat(C,D2))) ) ).

% prod_cases4
tff(fact_31_prod__cases3,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: heap_e7401611519738050253t_unit,C: set_nat] : ( Y != produc6017135503889078682et_nat(A3,produc7507926704131184380et_nat(B3,C)) ) ).

% prod_cases3
tff(fact_32_prod__cases3,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: fun_Pr6460944699146269935_nat_o,C: produc3658429121746597890et_nat] : ( Y != produc7291855603632349880et_nat(A3,produc6017135503889078682et_nat(B3,C)) ) ).

% prod_cases3
tff(fact_33_Pair__inject,axiom,
    ! [A: uint32,B: uint32,A2: uint32,B2: uint32] :
      ( ( produc1400373151660368625uint32(A,B) = produc1400373151660368625uint32(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_34_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_35_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( product_Pair_int_int(A,B) = product_Pair_int_int(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_36_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc3658429121746597890et_nat] :
      ( ( produc6017135503889078682et_nat(A,B) = produc6017135503889078682et_nat(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_37_Pair__inject,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,A2: fun_Pr6460944699146269935_nat_o,B2: produc8799533983703829792et_nat] :
      ( ( produc7291855603632349880et_nat(A,B) = produc7291855603632349880et_nat(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_38_prod__cases,axiom,
    ! [P2: fun_Pr1754293368803457890nt32_o,P3: produc827990862158126777uint32] :
      ( ! [A3: uint32,B3: uint32] : aa_Pro111665056643909595nt32_o(P2,produc1400373151660368625uint32(A3,B3))
     => aa_Pro111665056643909595nt32_o(P2,P3) ) ).

% prod_cases
tff(fact_39_prod__cases,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,P3: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : aa_Pro6150637604445964947_nat_o(P2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3))
     => aa_Pro6150637604445964947_nat_o(P2,P3) ) ).

% prod_cases
tff(fact_40_prod__cases,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,P3: product_prod_int_int] :
      ( ! [A3: int,B3: int] : aa_Pro4592068669816069595_int_o(P2,product_Pair_int_int(A3,B3))
     => aa_Pro4592068669816069595_int_o(P2,P3) ) ).

% prod_cases
tff(fact_41_prod__cases,axiom,
    ! [P2: fun_Pr2620521676128474833_nat_o,P3: produc8799533983703829792et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] : aa_Pro5643009357358181178_nat_o(P2,produc6017135503889078682et_nat(A3,B3))
     => aa_Pro5643009357358181178_nat_o(P2,P3) ) ).

% prod_cases
tff(fact_42_prod__cases,axiom,
    ! [P2: fun_Pr1227177680593602867_nat_o,P3: produc3112576180687729086et_nat] :
      ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] : aa_Pro2671238023309943196_nat_o(P2,produc7291855603632349880et_nat(A3,B3))
     => aa_Pro2671238023309943196_nat_o(P2,P3) ) ).

% prod_cases
tff(fact_43_surj__pair,axiom,
    ! [P3: produc827990862158126777uint32] :
    ? [X: uint32,Y3: uint32] : ( P3 = produc1400373151660368625uint32(X,Y3) ) ).

% surj_pair
tff(fact_44_surj__pair,axiom,
    ! [P3: product_prod_nat_nat] :
    ? [X: nat,Y3: nat] : ( P3 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3) ) ).

% surj_pair
tff(fact_45_surj__pair,axiom,
    ! [P3: product_prod_int_int] :
    ? [X: int,Y3: int] : ( P3 = product_Pair_int_int(X,Y3) ) ).

% surj_pair
tff(fact_46_surj__pair,axiom,
    ! [P3: produc8799533983703829792et_nat] :
    ? [X: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] : ( P3 = produc6017135503889078682et_nat(X,Y3) ) ).

% surj_pair
tff(fact_47_surj__pair,axiom,
    ! [P3: produc3112576180687729086et_nat] :
    ? [X: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] : ( P3 = produc7291855603632349880et_nat(X,Y3) ) ).

% surj_pair
tff(fact_48_old_Oprod_Oexhaust,axiom,
    ! [Y: produc827990862158126777uint32] :
      ~ ! [A3: uint32,B3: uint32] : ( Y != produc1400373151660368625uint32(A3,B3) ) ).

% old.prod.exhaust
tff(fact_49_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] : ( Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) ) ).

% old.prod.exhaust
tff(fact_50_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : ( Y != product_Pair_int_int(A3,B3) ) ).

% old.prod.exhaust
tff(fact_51_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8799533983703829792et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] : ( Y != produc6017135503889078682et_nat(A3,B3) ) ).

% old.prod.exhaust
tff(fact_52_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3112576180687729086et_nat] :
      ~ ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] : ( Y != produc7291855603632349880et_nat(A3,B3) ) ).

% old.prod.exhaust
tff(fact_53_one__assn__raw_Ocases,axiom,
    ! [Xa: produc3658429121746597890et_nat] :
      ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] : ( Xa != produc7507926704131184380et_nat(H,As) ) ).

% one_assn_raw.cases
tff(fact_54_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(T)) ) ).

% set_vebt_finite
tff(fact_55_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),collect_VEBT_VEBT(P2))
    <=> aa_VEBT_VEBT_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_56_mem__Collect__eq,axiom,
    ! [A: real,P2: fun_real_o] :
      ( aa_set_real_o(member_real(A),collect_real(P2))
    <=> aa_real_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_57_mem__Collect__eq,axiom,
    ! [A: complex,P2: fun_complex_o] :
      ( aa_set_complex_o(member_complex(A),collect_complex(P2))
    <=> aa_complex_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_58_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(P2))
    <=> aa_list_nat_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_59_mem__Collect__eq,axiom,
    ! [A: set_nat,P2: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(A),collect_set_nat(P2))
    <=> aa_set_nat_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_60_mem__Collect__eq,axiom,
    ! [A: nat,P2: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P2))
    <=> aa_nat_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_61_mem__Collect__eq,axiom,
    ! [A: int,P2: fun_int_o] :
      ( aa_set_int_o(member_int(A),collect_int(P2))
    <=> aa_int_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_62_Collect__mem__eq,axiom,
    ! [A4: set_VEBT_VEBT] : ( collect_VEBT_VEBT(aTP_Lamm_a(A4)) = A4 ) ).

% Collect_mem_eq
tff(fact_63_Collect__mem__eq,axiom,
    ! [A4: set_real] : ( collect_real(aTP_Lamm_aa(A4)) = A4 ) ).

% Collect_mem_eq
tff(fact_64_Collect__mem__eq,axiom,
    ! [A4: set_complex] : ( collect_complex(aTP_Lamm_ab(A4)) = A4 ) ).

% Collect_mem_eq
tff(fact_65_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] : ( collect_list_nat(aTP_Lamm_ac(A4)) = A4 ) ).

% Collect_mem_eq
tff(fact_66_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] : ( collect_set_nat(aTP_Lamm_ad(A4)) = A4 ) ).

% Collect_mem_eq
tff(fact_67_Collect__mem__eq,axiom,
    ! [A4: set_nat] : ( collect_nat(aTP_Lamm_ae(A4)) = A4 ) ).

% Collect_mem_eq
tff(fact_68_Collect__mem__eq,axiom,
    ! [A4: set_int] : ( collect_int(aTP_Lamm_af(A4)) = A4 ) ).

% Collect_mem_eq
tff(fact_69_Collect__cong,axiom,
    ! [P2: fun_complex_o,Q2: fun_complex_o] :
      ( ! [X: complex] :
          ( aa_complex_o(P2,X)
        <=> aa_complex_o(Q2,X) )
     => ( collect_complex(P2) = collect_complex(Q2) ) ) ).

% Collect_cong
tff(fact_70_Collect__cong,axiom,
    ! [P2: fun_list_nat_o,Q2: fun_list_nat_o] :
      ( ! [X: list_nat] :
          ( aa_list_nat_o(P2,X)
        <=> aa_list_nat_o(Q2,X) )
     => ( collect_list_nat(P2) = collect_list_nat(Q2) ) ) ).

% Collect_cong
tff(fact_71_Collect__cong,axiom,
    ! [P2: fun_set_nat_o,Q2: fun_set_nat_o] :
      ( ! [X: set_nat] :
          ( aa_set_nat_o(P2,X)
        <=> aa_set_nat_o(Q2,X) )
     => ( collect_set_nat(P2) = collect_set_nat(Q2) ) ) ).

% Collect_cong
tff(fact_72_Collect__cong,axiom,
    ! [P2: fun_nat_o,Q2: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
        <=> aa_nat_o(Q2,X) )
     => ( collect_nat(P2) = collect_nat(Q2) ) ) ).

% Collect_cong
tff(fact_73_Collect__cong,axiom,
    ! [P2: fun_int_o,Q2: fun_int_o] :
      ( ! [X: int] :
          ( aa_int_o(P2,X)
        <=> aa_int_o(Q2,X) )
     => ( collect_int(P2) = collect_int(Q2) ) ) ).

% Collect_cong
tff(fact_74_VEBT__internal_OperInsTrans_Ointros_I1_J,axiom,
    ! [T: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,T),T) ).

% VEBT_internal.perInsTrans.intros(1)
tff(fact_75_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_76_valid__0__not,axiom,
    ! [T: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),zero_zero_nat) ).

% valid_0_not
tff(fact_77_min__Null__member,axiom,
    ! [T: vEBT_VEBT,Xa: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xa) ) ).

% min_Null_member
tff(fact_78_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),X_1) ) ).

% not_min_Null_member
tff(fact_79_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Na)
     => ( Deg = Na ) ) ).

% deg_deg_n
tff(fact_80_ssubst__Pair__rhs,axiom,
    ! [R: uint32,Sa: uint32,R2: set_Pr1773385645901665561uint32,S: uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(R,Sa),R2)
     => ( ( S = Sa )
       => member8027108493173000802uint32(produc1400373151660368625uint32(R,S),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_81_ssubst__Pair__rhs,axiom,
    ! [R: nat,Sa: nat,R2: set_Pr1261947904930325089at_nat,S: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R),Sa),R2)
     => ( ( S = Sa )
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R),S),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_82_ssubst__Pair__rhs,axiom,
    ! [R: int,Sa: int,R2: set_Pr958786334691620121nt_int,S: int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(R,Sa),R2)
     => ( ( S = Sa )
       => member5262025264175285858nt_int(product_Pair_int_int(R,S),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_83_ssubst__Pair__rhs,axiom,
    ! [R: fun_Pr6460944699146269935_nat_o,Sa: produc3658429121746597890et_nat,R2: set_Pr1312406793078187350et_nat,S: produc3658429121746597890et_nat] :
      ( member4513763875858039863et_nat(produc6017135503889078682et_nat(R,Sa),R2)
     => ( ( S = Sa )
       => member4513763875858039863et_nat(produc6017135503889078682et_nat(R,S),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_84_ssubst__Pair__rhs,axiom,
    ! [R: fun_Pr6460944699146269935_nat_o,Sa: produc8799533983703829792et_nat,R2: set_Pr4040394263813967220et_nat,S: produc8799533983703829792et_nat] :
      ( member9174161442226468181et_nat(produc7291855603632349880et_nat(R,Sa),R2)
     => ( ( S = Sa )
       => member9174161442226468181et_nat(produc7291855603632349880et_nat(R,S),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_85_bex2I,axiom,
    ! [A: uint32,B: uint32,S2: set_Pr1773385645901665561uint32,P2: fun_ui3478071520421546738nt32_o] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),S2)
     => ( ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),S2)
         => aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,A),B) )
       => ? [A3: uint32,B3: uint32] :
            ( member8027108493173000802uint32(produc1400373151660368625uint32(A3,B3),S2)
            & aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,A3),B3) ) ) ) ).

% bex2I
tff(fact_86_bex2I,axiom,
    ! [A: nat,B: nat,S2: set_Pr1261947904930325089at_nat,P2: fun_nat_fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),S2)
     => ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),S2)
         => aa_nat_o(aa_nat_fun_nat_o(P2,A),B) )
       => ? [A3: nat,B3: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3),S2)
            & aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) ) ) ) ).

% bex2I
tff(fact_87_bex2I,axiom,
    ! [A: int,B: int,S2: set_Pr958786334691620121nt_int,P2: fun_int_fun_int_o] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),S2)
     => ( ( member5262025264175285858nt_int(product_Pair_int_int(A,B),S2)
         => aa_int_o(aa_int_fun_int_o(P2,A),B) )
       => ? [A3: int,B3: int] :
            ( member5262025264175285858nt_int(product_Pair_int_int(A3,B3),S2)
            & aa_int_o(aa_int_fun_int_o(P2,A3),B3) ) ) ) ).

% bex2I
tff(fact_88_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc3658429121746597890et_nat,S2: set_Pr1312406793078187350et_nat,P2: fun_fu8419763957888848259_nat_o] :
      ( member4513763875858039863et_nat(produc6017135503889078682et_nat(A,B),S2)
     => ( ( member4513763875858039863et_nat(produc6017135503889078682et_nat(A,B),S2)
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P2,A),B) )
       => ? [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
            ( member4513763875858039863et_nat(produc6017135503889078682et_nat(A3,B3),S2)
            & aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P2,A3),B3) ) ) ) ).

% bex2I
tff(fact_89_bex2I,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B: produc8799533983703829792et_nat,S2: set_Pr4040394263813967220et_nat,P2: fun_fu2611741329380808165_nat_o] :
      ( member9174161442226468181et_nat(produc7291855603632349880et_nat(A,B),S2)
     => ( ( member9174161442226468181et_nat(produc7291855603632349880et_nat(A,B),S2)
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P2,A),B) )
       => ? [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
            ( member9174161442226468181et_nat(produc7291855603632349880et_nat(A3,B3),S2)
            & aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P2,A3),B3) ) ) ) ).

% bex2I
tff(fact_90_set__vebt__minNull,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
      <=> ( vEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% set_vebt_minNull
tff(fact_91_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Y11,Y12,Y13),Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_92_deg__not__0,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% deg_not_0
tff(fact_93_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] : ( Tree = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Na)),TreeList2),S3) ) ) ).

% deg_SUcn_Node
tff(fact_94_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_95_divides__aux__eq,axiom,
    ! [Q3: nat,R: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R))
    <=> ( R = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_96_divides__aux__eq,axiom,
    ! [Q3: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,product_Pair_int_int(Q3,R))
    <=> ( R = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_97_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_98_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X22)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_99_not__gr__zero,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_100_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X3) ) ) ) ).

% succ_none_empty
tff(fact_101_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),A) ) ) ) ).

% pred_none_empty
tff(fact_102_memb__imp__not__empty,axiom,
    ! [Xa: vEBT_VEBT,S2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),S2)
     => ( S2 != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
tff(fact_103_memb__imp__not__empty,axiom,
    ! [Xa: real,S2: set_real] :
      ( aa_set_real_o(member_real(Xa),S2)
     => ( S2 != bot_bot_set_real ) ) ).

% memb_imp_not_empty
tff(fact_104_memb__imp__not__empty,axiom,
    ! [Xa: set_nat,S2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),S2)
     => ( S2 != bot_bot_set_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_105_memb__imp__not__empty,axiom,
    ! [Xa: nat,S2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),S2)
     => ( S2 != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_106_memb__imp__not__empty,axiom,
    ! [Xa: int,S2: set_int] :
      ( aa_set_int_o(member_int(Xa),S2)
     => ( S2 != bot_bot_set_int ) ) ).

% memb_imp_not_empty
tff(fact_107_memb__imp__not__empty,axiom,
    ! [Xa: $o,S2: set_o] :
      ( aa_set_o_o(member_o((Xa)),S2)
     => ( S2 != bot_bot_set_o ) ) ).

% memb_imp_not_empty
tff(fact_108_set__notEmptyE,axiom,
    ! [S2: set_VEBT_VEBT] :
      ( ( S2 != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S2) ) ).

% set_notEmptyE
tff(fact_109_set__notEmptyE,axiom,
    ! [S2: set_real] :
      ( ( S2 != bot_bot_set_real )
     => ~ ! [X: real] : ~ aa_set_real_o(member_real(X),S2) ) ).

% set_notEmptyE
tff(fact_110_set__notEmptyE,axiom,
    ! [S2: set_set_nat] :
      ( ( S2 != bot_bot_set_set_nat )
     => ~ ! [X: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X),S2) ) ).

% set_notEmptyE
tff(fact_111_set__notEmptyE,axiom,
    ! [S2: set_nat] :
      ( ( S2 != bot_bot_set_nat )
     => ~ ! [X: nat] : ~ aa_set_nat_o(member_nat(X),S2) ) ).

% set_notEmptyE
tff(fact_112_set__notEmptyE,axiom,
    ! [S2: set_int] :
      ( ( S2 != bot_bot_set_int )
     => ~ ! [X: int] : ~ aa_set_int_o(member_int(X),S2) ) ).

% set_notEmptyE
tff(fact_113_set__notEmptyE,axiom,
    ! [S2: set_o] :
      ( ( S2 != bot_bot_set_o )
     => ~ ! [X: $o] : ~ aa_set_o_o(member_o((X)),S2) ) ).

% set_notEmptyE
tff(fact_114_gr__zeroI,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr_zeroI
tff(fact_115_not__less__zero,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less_zero
tff(fact_116_gr__implies__not__zero,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_117_zero__less__iff__neq__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_118_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] : ( aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) != vEBT_Leaf((X21),(X22)) ) ).

% VEBT.distinct(1)
tff(fact_119_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(X112,X122,X132),X142) )
     => ~ ! [X212: $o,X222: $o] : ( Y != vEBT_Leaf((X212),(X222)) ) ) ).

% VEBT.exhaust
tff(fact_120_finite_OemptyI,axiom,
    aa_set_complex_o(finite3207457112153483333omplex,bot_bot_set_complex) ).

% finite.emptyI
tff(fact_121_finite_OemptyI,axiom,
    aa_set5643387332196575111eger_o(finite6017078050557962740nteger,bot_bo3990330152332043303nteger) ).

% finite.emptyI
tff(fact_122_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_123_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_124_finite_OemptyI,axiom,
    aa_set_o_o(finite_finite_o,bot_bot_set_o) ).

% finite.emptyI
tff(fact_125_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_126_infinite__imp__nonempty,axiom,
    ! [S2: set_Code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( S2 != bot_bo3990330152332043303nteger ) ) ).

% infinite_imp_nonempty
tff(fact_127_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_128_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_129_infinite__imp__nonempty,axiom,
    ! [S2: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,S2)
     => ( S2 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
tff(fact_130_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X: nat] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X) )
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw),Ux) )
       => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3),X) ) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_131_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_132_mod__h__bot__indep,axiom,
    ! [P2: assn,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P2),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(rep_assn(P2),produc7507926704131184380et_nat(H3,bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_133_zero__reorient,axiom,
    ! [Xa: uint32] :
      ( ( zero_zero_uint32 = Xa )
    <=> ( Xa = zero_zero_uint32 ) ) ).

% zero_reorient
tff(fact_134_zero__reorient,axiom,
    ! [Xa: real] :
      ( ( zero_zero_real = Xa )
    <=> ( Xa = zero_zero_real ) ) ).

% zero_reorient
tff(fact_135_zero__reorient,axiom,
    ! [Xa: rat] :
      ( ( zero_zero_rat = Xa )
    <=> ( Xa = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_136_zero__reorient,axiom,
    ! [Xa: nat] :
      ( ( zero_zero_nat = Xa )
    <=> ( Xa = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_137_zero__reorient,axiom,
    ! [Xa: int] :
      ( ( zero_zero_int = Xa )
    <=> ( Xa = zero_zero_int ) ) ).

% zero_reorient
tff(fact_138_zero__less__Suc,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Na)) ).

% zero_less_Suc
tff(fact_139_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_140_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_141_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% neq0_conv
tff(fact_142_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_143_lessI,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Na)) ).

% lessI
tff(fact_144_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) ) ).

% Suc_mono
tff(fact_145_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_eq
tff(fact_146_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf((A),$false) ) ).

% vebt_delete.simps(2)
tff(fact_147_invar__vebt__buildup,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% invar_vebt_buildup
tff(fact_148_buildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na)),Na) ) ).

% buildup_gives_valid
tff(fact_149_T__vebt__buildupi__gq__0,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% T_vebt_buildupi_gq_0
tff(fact_150_set__vebt__buildup,axiom,
    ! [I: nat] : ( vEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_buildup,I)) = bot_bot_set_nat ) ).

% set_vebt_buildup
tff(fact_151_buildup__gives__empty,axiom,
    ! [Na: nat] : ( vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na)) = bot_bot_set_nat ) ).

% buildup_gives_empty
tff(fact_152_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_153_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_154_obtain__set__pred,axiom,
    ! [Z: nat,Xa: nat,A4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xa)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A4),Z)
       => ( aa_set_nat_o(finite_finite_nat,A4)
         => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A4),Xa),X_1) ) ) ) ).

% obtain_set_pred
tff(fact_155_obtain__set__succ,axiom,
    ! [Xa: nat,Z: nat,A4: set_nat,B4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A4),Z)
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => ( ( A4 = B4 )
           => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A4),Xa),X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_156_finite__psubset__induct,axiom,
    ! [A4: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [A5: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A5)
           => ( ! [B5: set_nat] :
                  ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B5),A5)
                 => aa_set_nat_o(P2,B5) )
             => aa_set_nat_o(P2,A5) ) )
       => aa_set_nat_o(P2,A4) ) ) ).

% finite_psubset_induct
tff(fact_157_finite__psubset__induct,axiom,
    ! [A4: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [A5: set_int] :
            ( aa_set_int_o(finite_finite_int,A5)
           => ( ! [B5: set_int] :
                  ( aa_set_int_o(ord_less_set_int(B5),A5)
                 => aa_set_int_o(P2,B5) )
             => aa_set_int_o(P2,A5) ) )
       => aa_set_int_o(P2,A4) ) ) ).

% finite_psubset_induct
tff(fact_158_finite__psubset__induct,axiom,
    ! [A4: set_complex,P2: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [A5: set_complex] :
            ( aa_set_complex_o(finite3207457112153483333omplex,A5)
           => ( ! [B5: set_complex] :
                  ( aa_set_complex_o(ord_less_set_complex(B5),A5)
                 => aa_set_complex_o(P2,B5) )
             => aa_set_complex_o(P2,A5) ) )
       => aa_set_complex_o(P2,A4) ) ) ).

% finite_psubset_induct
tff(fact_159_finite__psubset__induct,axiom,
    ! [A4: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [A5: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5)
           => ( ! [B5: set_Code_integer] :
                  ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(B5),A5)
                 => aa_set5643387332196575111eger_o(P2,B5) )
             => aa_set5643387332196575111eger_o(P2,A5) ) )
       => aa_set5643387332196575111eger_o(P2,A4) ) ) ).

% finite_psubset_induct
tff(fact_160_vebt__buildup_Osimps_I1_J,axiom,
    aa_nat_VEBT_VEBT(vEBT_vebt_buildup,zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_161_n__not__Suc__n,axiom,
    ! [Na: nat] : ( Na != aa_nat_nat(suc,Na) ) ).

% n_not_Suc_n
tff(fact_162_Suc__inject,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xa) = aa_nat_nat(suc,Y) )
     => ( Xa = Y ) ) ).

% Suc_inject
tff(fact_163_linorder__neqE__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_164_infinite__descent,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P2,N)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & ~ aa_nat_o(P2,M2) ) )
     => aa_nat_o(P2,Na) ) ).

% infinite_descent
tff(fact_165_nat__less__induct,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
             => aa_nat_o(P2,M2) )
         => aa_nat_o(P2,N) )
     => aa_nat_o(P2,Na) ) ).

% nat_less_induct
tff(fact_166_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_irrefl_nat
tff(fact_167_less__not__refl3,axiom,
    ! [Sa: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Sa),T)
     => ( Sa != T ) ) ).

% less_not_refl3
tff(fact_168_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_169_less__not__refl,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_not_refl
tff(fact_170_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% nat_neq_iff
tff(fact_171_vebt__buildup_Osimps_I2_J,axiom,
    aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_172_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ? [M3: nat] : ( Na = aa_nat_nat(suc,M3) ) ) ).

% not0_implies_Suc
tff(fact_173_Zero__not__Suc,axiom,
    ! [M: nat] : ( zero_zero_nat != aa_nat_nat(suc,M) ) ).

% Zero_not_Suc
tff(fact_174_Zero__neq__Suc,axiom,
    ! [M: nat] : ( zero_zero_nat != aa_nat_nat(suc,M) ) ).

% Zero_neq_Suc
tff(fact_175_Suc__neq__Zero,axiom,
    ! [M: nat] : ( aa_nat_nat(suc,M) != zero_zero_nat ) ).

% Suc_neq_Zero
tff(fact_176_zero__induct,axiom,
    ! [P2: fun_nat_o,K: nat] :
      ( aa_nat_o(P2,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P2,aa_nat_nat(suc,N))
           => aa_nat_o(P2,N) )
       => aa_nat_o(P2,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_177_diff__induct,axiom,
    ! [P2: fun_nat_fun_nat_o,M: nat,Na: nat] :
      ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,X),zero_zero_nat)
     => ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,zero_zero_nat),aa_nat_nat(suc,Y3))
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P2,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(P2,aa_nat_nat(suc,X)),aa_nat_nat(suc,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,M),Na) ) ) ) ).

% diff_induct
tff(fact_178_nat__induct,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P2,N)
           => aa_nat_o(P2,aa_nat_nat(suc,N)) )
       => aa_nat_o(P2,Na) ) ) ).

% nat_induct
tff(fact_179_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : ( Y != aa_nat_nat(suc,Nat3) ) ) ).

% old.nat.exhaust
tff(fact_180_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_181_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : ( zero_zero_nat != aa_nat_nat(suc,Nat2) ) ).

% old.nat.distinct(1)
tff(fact_182_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : ( aa_nat_nat(suc,Nat2) != zero_zero_nat ) ).

% old.nat.distinct(2)
tff(fact_183_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : ( zero_zero_nat != aa_nat_nat(suc,X2) ) ).

% nat.distinct(1)
tff(fact_184_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_185_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P2,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P2,aa_nat_nat(suc,I2))
               => aa_nat_o(P2,I2) ) )
         => aa_nat_o(P2,I) ) ) ) ).

% strict_inc_induct
tff(fact_186_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P2: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P2,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P2,J2),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P2,I2),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_187_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_188_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_SucD
tff(fact_189_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_190_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M)
    <=> ? [M4: nat] :
          ( ( M = aa_nat_nat(suc,M4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M4) ) ) ).

% Suc_less_eq2
tff(fact_191_Nat_OAll__less__Suc,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
         => aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,Na)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
           => aa_nat_o(P2,I3) ) ) ) ).

% Nat.All_less_Suc
tff(fact_192_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_193_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_194_Ex__less__Suc,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
          & aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,Na)
        | ? [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
            & aa_nat_o(P2,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_195_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% less_SucI
tff(fact_196_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_197_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( ( aa_nat_nat(suc,M) != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_198_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_199_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_lessD
tff(fact_200_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_201_infinite__descent0,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P2,N)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P2,M2) ) ) )
       => aa_nat_o(P2,Na) ) ) ).

% infinite_descent0
tff(fact_202_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_203_less__zeroE,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_zeroE
tff(fact_204_not__less0,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less0
tff(fact_205_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr0
tff(fact_206_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr0I
tff(fact_207_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_208_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Na: nat,M: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_209_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Na: nat,M: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_210_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Na: nat,M: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Na)),aa_nat_num(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_211_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Na: nat,M: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_212_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Na: nat,M: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_213_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_214_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_215_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Na)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_216_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_217_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_218_Ex__less__Suc2,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
          & aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,zero_zero_nat)
        | ? [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
            & aa_nat_o(P2,aa_nat_nat(suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_219_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ? [M5: nat] : ( Na = aa_nat_nat(suc,M5) ) ) ).

% gr0_conv_Suc
tff(fact_220_All__less__Suc2,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
         => aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,zero_zero_nat)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
           => aa_nat_o(P2,aa_nat_nat(suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_221_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ? [M3: nat] : ( Na = aa_nat_nat(suc,M3) ) ) ).

% gr0_implies_Suc
tff(fact_222_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_223_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Na: nat] : ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Na))) = vEBT_Leaf((A),(B)) ) ).

% vebt_delete.simps(3)
tff(fact_224_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(vEBT_Leaf((A),(B))),zero_zero_nat) = vEBT_Leaf($false,(B)) ) ).

% vebt_delete.simps(1)
tff(fact_225_succ__member,axiom,
    ! [T: vEBT_VEBT,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),Xa),Y)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        & ! [Z2: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Z2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z2) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z2) ) ) ) ).

% succ_member
tff(fact_226_pred__member,axiom,
    ! [T: vEBT_VEBT,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),Xa),Y)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
        & ! [Z2: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Z2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),Xa) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),Y) ) ) ) ).

% pred_member
tff(fact_227_buildup__nothing__in__leaf,axiom,
    ! [Na: nat,Xa: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na)),Xa) ).

% buildup_nothing_in_leaf
tff(fact_228_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_229_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,zero_zero_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_230_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X: nat] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X) )
     => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2),X) ) ) ).

% VEBT_internal.insert'.cases
tff(fact_231_forall__finite_I3_J,axiom,
    ! [Xa: nat,P2: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,aa_nat_nat(suc,Xa)))
         => aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,zero_zero_nat)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Xa))
           => aa_nat_o(P2,aa_nat_nat(suc,I3)) ) ) ) ).

% forall_finite(3)
tff(fact_232_forall__finite_I2_J,axiom,
    ! [P2: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,zero_zero_nat))
         => aa_nat_o(P2,I3) )
    <=> aa_nat_o(P2,zero_zero_nat) ) ).

% forall_finite(2)
tff(fact_233_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xa: nat,P2: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Xa))
         => aa_nat_o(P2,I3) )
    <=> ( aa_nat_o(P2,zero_zero_nat)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Xa)
           => aa_nat_o(P2,aa_nat_nat(suc,I3)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_234_infinite__growing,axiom,
    ! [X4: set_Code_integer] :
      ( ( X4 != bot_bo3990330152332043303nteger )
     => ( ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),X4)
           => ? [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),X4)
                & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Xa2) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X4) ) ) ).

% infinite_growing
tff(fact_235_infinite__growing,axiom,
    ! [X4: set_o] :
      ( ( X4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),X4)
           => ? [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),X4)
                & aa_o_o(aa_o_fun_o_o(ord_less_o,(X)),(Xa2)) ) )
       => ~ aa_set_o_o(finite_finite_o,X4) ) ) ).

% infinite_growing
tff(fact_236_infinite__growing,axiom,
    ! [X4: set_real] :
      ( ( X4 != bot_bot_set_real )
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),X4)
           => ? [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),X4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X),Xa2) ) )
       => ~ aa_set_real_o(finite_finite_real,X4) ) ) ).

% infinite_growing
tff(fact_237_infinite__growing,axiom,
    ! [X4: set_rat] :
      ( ( X4 != bot_bot_set_rat )
     => ( ! [X: rat] :
            ( aa_set_rat_o(member_rat(X),X4)
           => ? [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),X4)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Xa2) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X4) ) ) ).

% infinite_growing
tff(fact_238_infinite__growing,axiom,
    ! [X4: set_num] :
      ( ( X4 != bot_bot_set_num )
     => ( ! [X: num] :
            ( aa_set_num_o(member_num(X),X4)
           => ? [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),X4)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X),Xa2) ) )
       => ~ aa_set_num_o(finite_finite_num,X4) ) ) ).

% infinite_growing
tff(fact_239_infinite__growing,axiom,
    ! [X4: set_nat] :
      ( ( X4 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),X4)
           => ? [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),X4)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa2) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X4) ) ) ).

% infinite_growing
tff(fact_240_infinite__growing,axiom,
    ! [X4: set_int] :
      ( ( X4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),X4)
           => ? [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),X4)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa2) ) )
       => ~ aa_set_int_o(finite_finite_int,X4) ) ) ).

% infinite_growing
tff(fact_241_ex__min__if__finite,axiom,
    ! [S2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ? [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),S2)
            & ~ ? [Xa2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),S2)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_242_ex__min__if__finite,axiom,
    ! [S2: set_o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ( S2 != bot_bot_set_o )
       => ? [X: $o] :
            ( aa_set_o_o(member_o((X)),S2)
            & ~ ? [Xa2: $o] :
                  ( aa_set_o_o(member_o((Xa2)),S2)
                  & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa2)),(X)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_243_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( ( S2 != bot_bot_set_real )
       => ? [X: real] :
            ( aa_set_real_o(member_real(X),S2)
            & ~ ? [Xa2: real] :
                  ( aa_set_real_o(member_real(Xa2),S2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_244_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S2)
     => ( ( S2 != bot_bot_set_rat )
       => ? [X: rat] :
            ( aa_set_rat_o(member_rat(X),S2)
            & ~ ? [Xa2: rat] :
                  ( aa_set_rat_o(member_rat(Xa2),S2)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_245_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( aa_set_num_o(finite_finite_num,S2)
     => ( ( S2 != bot_bot_set_num )
       => ? [X: num] :
            ( aa_set_num_o(member_num(X),S2)
            & ~ ? [Xa2: num] :
                  ( aa_set_num_o(member_num(Xa2),S2)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_246_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),S2)
            & ~ ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),S2)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_247_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),S2)
            & ~ ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),S2)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa2),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_248_empty__Collect__eq,axiom,
    ! [P2: fun_complex_o] :
      ( ( bot_bot_set_complex = collect_complex(P2) )
    <=> ! [X5: complex] : ~ aa_complex_o(P2,X5) ) ).

% empty_Collect_eq
tff(fact_249_empty__Collect__eq,axiom,
    ! [P2: fun_list_nat_o] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P2) )
    <=> ! [X5: list_nat] : ~ aa_list_nat_o(P2,X5) ) ).

% empty_Collect_eq
tff(fact_250_empty__Collect__eq,axiom,
    ! [P2: fun_set_nat_o] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P2) )
    <=> ! [X5: set_nat] : ~ aa_set_nat_o(P2,X5) ) ).

% empty_Collect_eq
tff(fact_251_empty__Collect__eq,axiom,
    ! [P2: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(P2) )
    <=> ! [X5: nat] : ~ aa_nat_o(P2,X5) ) ).

% empty_Collect_eq
tff(fact_252_empty__Collect__eq,axiom,
    ! [P2: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(P2) )
    <=> ! [X5: int] : ~ aa_int_o(P2,X5) ) ).

% empty_Collect_eq
tff(fact_253_empty__Collect__eq,axiom,
    ! [P2: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(P2) )
    <=> ! [X5: $o] : ~ aa_o_o(P2,(X5)) ) ).

% empty_Collect_eq
tff(fact_254_max__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xa)
    <=> ( aa_set_nat_o(member_nat(Xa),Xs)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa) ) ) ) ).

% max_in_set_def
tff(fact_255_min__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xa)
    <=> ( aa_set_nat_o(member_nat(Xa),Xs)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X5) ) ) ) ).

% min_in_set_def
tff(fact_256_empty__iff,axiom,
    ! [C2: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),bot_bo8194388402131092736T_VEBT) ).

% empty_iff
tff(fact_257_empty__iff,axiom,
    ! [C2: real] : ~ aa_set_real_o(member_real(C2),bot_bot_set_real) ).

% empty_iff
tff(fact_258_empty__iff,axiom,
    ! [C2: set_nat] : ~ aa_set_set_nat_o(member_set_nat(C2),bot_bot_set_set_nat) ).

% empty_iff
tff(fact_259_empty__iff,axiom,
    ! [C2: nat] : ~ aa_set_nat_o(member_nat(C2),bot_bot_set_nat) ).

% empty_iff
tff(fact_260_empty__iff,axiom,
    ! [C2: int] : ~ aa_set_int_o(member_int(C2),bot_bot_set_int) ).

% empty_iff
tff(fact_261_empty__iff,axiom,
    ! [C2: $o] : ~ aa_set_o_o(member_o((C2)),bot_bot_set_o) ).

% empty_iff
tff(fact_262_all__not__in__conv,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ! [X5: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),A4)
    <=> ( A4 = bot_bo8194388402131092736T_VEBT ) ) ).

% all_not_in_conv
tff(fact_263_all__not__in__conv,axiom,
    ! [A4: set_real] :
      ( ! [X5: real] : ~ aa_set_real_o(member_real(X5),A4)
    <=> ( A4 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_264_all__not__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ! [X5: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X5),A4)
    <=> ( A4 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_265_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ! [X5: nat] : ~ aa_set_nat_o(member_nat(X5),A4)
    <=> ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_266_all__not__in__conv,axiom,
    ! [A4: set_int] :
      ( ! [X5: int] : ~ aa_set_int_o(member_int(X5),A4)
    <=> ( A4 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_267_all__not__in__conv,axiom,
    ! [A4: set_o] :
      ( ! [X5: $o] : ~ aa_set_o_o(member_o((X5)),A4)
    <=> ( A4 = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_268_Collect__empty__eq,axiom,
    ! [P2: fun_complex_o] :
      ( ( collect_complex(P2) = bot_bot_set_complex )
    <=> ! [X5: complex] : ~ aa_complex_o(P2,X5) ) ).

% Collect_empty_eq
tff(fact_269_Collect__empty__eq,axiom,
    ! [P2: fun_list_nat_o] :
      ( ( collect_list_nat(P2) = bot_bot_set_list_nat )
    <=> ! [X5: list_nat] : ~ aa_list_nat_o(P2,X5) ) ).

% Collect_empty_eq
tff(fact_270_Collect__empty__eq,axiom,
    ! [P2: fun_set_nat_o] :
      ( ( collect_set_nat(P2) = bot_bot_set_set_nat )
    <=> ! [X5: set_nat] : ~ aa_set_nat_o(P2,X5) ) ).

% Collect_empty_eq
tff(fact_271_Collect__empty__eq,axiom,
    ! [P2: fun_nat_o] :
      ( ( collect_nat(P2) = bot_bot_set_nat )
    <=> ! [X5: nat] : ~ aa_nat_o(P2,X5) ) ).

% Collect_empty_eq
tff(fact_272_Collect__empty__eq,axiom,
    ! [P2: fun_int_o] :
      ( ( collect_int(P2) = bot_bot_set_int )
    <=> ! [X5: int] : ~ aa_int_o(P2,X5) ) ).

% Collect_empty_eq
tff(fact_273_Collect__empty__eq,axiom,
    ! [P2: fun_o_o] :
      ( ( collect_o(P2) = bot_bot_set_o )
    <=> ! [X5: $o] : ~ aa_o_o(P2,(X5)) ) ).

% Collect_empty_eq
tff(fact_274_le__zero__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_275_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% Suc_le_mono
tff(fact_276_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_277_le0,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% le0
tff(fact_278_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
     => ( aa_set_nat_o(member_nat(C2),A4)
       => aa_set_nat_o(member_nat(C2),B4) ) ) ).

% psubsetD
tff(fact_279_psubsetD,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A4),B4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A4)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B4) ) ) ).

% psubsetD
tff(fact_280_psubsetD,axiom,
    ! [A4: set_real,B4: set_real,C2: real] :
      ( aa_set_real_o(ord_less_set_real(A4),B4)
     => ( aa_set_real_o(member_real(C2),A4)
       => aa_set_real_o(member_real(C2),B4) ) ) ).

% psubsetD
tff(fact_281_psubsetD,axiom,
    ! [A4: set_int,B4: set_int,C2: int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => ( aa_set_int_o(member_int(C2),A4)
       => aa_set_int_o(member_int(C2),B4) ) ) ).

% psubsetD
tff(fact_282_psubsetD,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,C2: set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A4),B4)
     => ( aa_set_set_nat_o(member_set_nat(C2),A4)
       => aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% psubsetD
tff(fact_283_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P2,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P2,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => ? [X: nat] :
            ( aa_nat_o(P2,X)
            & ! [Y4: nat] :
                ( aa_nat_o(P2,Y4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_284_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% nat_le_linear
tff(fact_285_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_286_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% eq_imp_le
tff(fact_287_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_288_le__refl,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Na) ).

% le_refl
tff(fact_289_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int,D: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(ord_less_eq_set_int(B),C2)
       => ( ( C2 = D )
         => aa_set_int_o(ord_less_eq_set_int(A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_290_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ( C2 = D )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_291_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ( C2 = D )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_292_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ( C2 = D )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_293_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ( C2 = D )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_294_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N))),aa_nat_set_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_295_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_296_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_297_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_298_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_299_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,Na)),aa_nat_set_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_300_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_301_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Na)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_302_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_303_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Na)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_304_zero__le,axiom,
    ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa) ).

% zero_le
tff(fact_305_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(A),A4)
       => ? [X: real] :
            ( aa_set_real_o(member_real(X),A4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
            & ! [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_306_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(member_set_nat(A),A4)
       => ? [X: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X),A4)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A4)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_307_finite__has__maximal2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A4)
       => ? [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),X)
            & ! [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A4)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_308_finite__has__maximal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( member_set_int(A,A4)
       => ? [X: set_int] :
            ( member_set_int(X,A4)
            & aa_set_int_o(ord_less_eq_set_int(A),X)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A4)
               => ( aa_set_int_o(ord_less_eq_set_int(X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_309_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(A),A4)
       => ? [X: rat] :
            ( aa_set_rat_o(member_rat(X),A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_310_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(A),A4)
       => ? [X: num] :
            ( aa_set_num_o(member_num(X),A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_311_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_312_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_313_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(A),A4)
       => ? [X: real] :
            ( aa_set_real_o(member_real(X),A4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),A)
            & ! [Xa2: real] :
                ( aa_set_real_o(member_real(Xa2),A4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_314_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(member_set_nat(A),A4)
       => ? [X: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X),A4)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
            & ! [Xa2: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa2),A4)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_315_finite__has__minimal2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A4)
       => ? [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),A)
            & ! [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A4)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_316_finite__has__minimal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( member_set_int(A,A4)
       => ? [X: set_int] :
            ( member_set_int(X,A4)
            & aa_set_int_o(ord_less_eq_set_int(X),A)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A4)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_317_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(A),A4)
       => ? [X: rat] :
            ( aa_set_rat_o(member_rat(X),A4)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_318_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(A),A4)
       => ? [X: num] :
            ( aa_set_num_o(member_num(X),A4)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),A)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_319_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_320_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_321_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_leD
tff(fact_322_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_SucE
tff(fact_323_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_SucI
tff(fact_324_Suc__le__D,axiom,
    ! [Na: nat,M6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M6)
     => ? [M3: nat] : ( M6 = aa_nat_nat(suc,M3) ) ) ).

% Suc_le_D
tff(fact_325_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        | ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_326_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_327_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_328_full__nat__induct,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)
             => aa_nat_o(P2,M2) )
         => aa_nat_o(P2,N) )
     => aa_nat_o(P2,Na) ) ).

% full_nat_induct
tff(fact_329_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(P2,M)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
             => ( aa_nat_o(P2,N)
               => aa_nat_o(P2,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P2,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_330_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R2: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(R2,X),X)
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R2,X),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R2,Y3),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R2,X),Z3) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R2,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R2,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_331_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% less_eq_nat.simps(1)
tff(fact_332_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_333_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_334_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_335_exists__leI,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( ( ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),Na)
           => ~ aa_nat_o(P2,N3) )
       => aa_nat_o(P2,Na) )
     => ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),Na)
          & aa_nat_o(P2,N4) ) ) ).

% exists_leI
tff(fact_336_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_337_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_imp_le_nat
tff(fact_338_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_339_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_or_eq_imp_le
tff(fact_340_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ( M != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% le_neq_implies_less
tff(fact_341_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_342_finite__has__minimal,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
            & ! [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A4)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_343_finite__has__minimal,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ? [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa2)),(X))
                 => ( (X)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_344_finite__has__minimal,axiom,
    ! [A4: set_set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X: set_int] :
            ( member_set_int(X,A4)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A4)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_345_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X: rat] :
            ( aa_set_rat_o(member_rat(X),A4)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_346_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X: num] :
            ( aa_set_num_o(member_num(X),A4)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_347_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_348_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa2),X)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_349_finite__has__maximal,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
            & ! [Xa2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A4)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_350_finite__has__maximal,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ? [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
            & ! [Xa2: $o] :
                ( aa_set_o_o(member_o((Xa2)),A4)
               => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X)),(Xa2))
                 => ( (X)
                  <=> (Xa2) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_351_finite__has__maximal,axiom,
    ! [A4: set_set_int] :
      ( finite6197958912794628473et_int(A4)
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X: set_int] :
            ( member_set_int(X,A4)
            & ! [Xa2: set_int] :
                ( member_set_int(Xa2,A4)
               => ( aa_set_int_o(ord_less_eq_set_int(X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_352_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ? [X: rat] :
            ( aa_set_rat_o(member_rat(X),A4)
            & ! [Xa2: rat] :
                ( aa_set_rat_o(member_rat(Xa2),A4)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_353_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ? [X: num] :
            ( aa_set_num_o(member_num(X),A4)
            & ! [Xa2: num] :
                ( aa_set_num_o(member_num(Xa2),A4)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_354_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ! [Xa2: nat] :
                ( aa_set_nat_o(member_nat(Xa2),A4)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_355_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),A4)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2)
                 => ( X = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_356_nat__compl__induct,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N)
               => aa_nat_o(P2,Nn) )
           => aa_nat_o(P2,aa_nat_nat(suc,N)) )
       => aa_nat_o(P2,Na) ) ) ).

% nat_compl_induct
tff(fact_357_nat__compl__induct_H,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N)
               => aa_nat_o(P2,Nn) )
           => aa_nat_o(P2,aa_nat_nat(suc,N)) )
       => aa_nat_o(P2,Na) ) ) ).

% nat_compl_induct'
tff(fact_358_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = aa_nat_nat(suc,A) ) ) ).

% nat_in_between_eq(1)
tff(fact_359_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(suc,A)) )
    <=> ( B = A ) ) ).

% nat_in_between_eq(2)
tff(fact_360_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na) ) ).

% Suc_leI
tff(fact_361_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_eq
tff(fact_362_dec__induct,axiom,
    ! [I: nat,J: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P2,I)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P2,N)
                 => aa_nat_o(P2,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P2,J) ) ) ) ).

% dec_induct
tff(fact_363_inc__induct,axiom,
    ! [I: nat,J: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P2,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P2,aa_nat_nat(suc,N))
                 => aa_nat_o(P2,N) ) ) )
         => aa_nat_o(P2,I) ) ) ) ).

% inc_induct
tff(fact_364_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_lessD
tff(fact_365_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_366_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_Suc_eq_le
tff(fact_367_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_368_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_369_ex__least__nat__le,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( aa_nat_o(P2,Na)
     => ( ~ aa_nat_o(P2,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
               => ~ aa_nat_o(P2,I4) )
            & aa_nat_o(P2,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_370_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw2)),Ux2) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_371_ex__least__nat__less,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( aa_nat_o(P2,Na)
     => ( ~ aa_nat_o(P2,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K2)
               => ~ aa_nat_o(P2,I4) )
            & aa_nat_o(P2,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_372_emptyE,axiom,
    ! [A: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ).

% emptyE
tff(fact_373_emptyE,axiom,
    ! [A: real] : ~ aa_set_real_o(member_real(A),bot_bot_set_real) ).

% emptyE
tff(fact_374_emptyE,axiom,
    ! [A: set_nat] : ~ aa_set_set_nat_o(member_set_nat(A),bot_bot_set_set_nat) ).

% emptyE
tff(fact_375_emptyE,axiom,
    ! [A: nat] : ~ aa_set_nat_o(member_nat(A),bot_bot_set_nat) ).

% emptyE
tff(fact_376_emptyE,axiom,
    ! [A: int] : ~ aa_set_int_o(member_int(A),bot_bot_set_int) ).

% emptyE
tff(fact_377_emptyE,axiom,
    ! [A: $o] : ~ aa_set_o_o(member_o((A)),bot_bot_set_o) ).

% emptyE
tff(fact_378_equals0D,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( A4 = bot_bo8194388402131092736T_VEBT )
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4) ) ).

% equals0D
tff(fact_379_equals0D,axiom,
    ! [A4: set_real,A: real] :
      ( ( A4 = bot_bot_set_real )
     => ~ aa_set_real_o(member_real(A),A4) ) ).

% equals0D
tff(fact_380_equals0D,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( A4 = bot_bot_set_set_nat )
     => ~ aa_set_set_nat_o(member_set_nat(A),A4) ) ).

% equals0D
tff(fact_381_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A),A4) ) ).

% equals0D
tff(fact_382_equals0D,axiom,
    ! [A4: set_int,A: int] :
      ( ( A4 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A),A4) ) ).

% equals0D
tff(fact_383_equals0D,axiom,
    ! [A4: set_o,A: $o] :
      ( ( A4 = bot_bot_set_o )
     => ~ aa_set_o_o(member_o((A)),A4) ) ).

% equals0D
tff(fact_384_equals0I,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ! [Y3: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y3),A4)
     => ( A4 = bot_bo8194388402131092736T_VEBT ) ) ).

% equals0I
tff(fact_385_equals0I,axiom,
    ! [A4: set_real] :
      ( ! [Y3: real] : ~ aa_set_real_o(member_real(Y3),A4)
     => ( A4 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_386_equals0I,axiom,
    ! [A4: set_set_nat] :
      ( ! [Y3: set_nat] : ~ aa_set_set_nat_o(member_set_nat(Y3),A4)
     => ( A4 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_387_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] : ~ aa_set_nat_o(member_nat(Y3),A4)
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_388_equals0I,axiom,
    ! [A4: set_int] :
      ( ! [Y3: int] : ~ aa_set_int_o(member_int(Y3),A4)
     => ( A4 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_389_equals0I,axiom,
    ! [A4: set_o] :
      ( ! [Y3: $o] : ~ aa_set_o_o(member_o((Y3)),A4)
     => ( A4 = bot_bot_set_o ) ) ).

% equals0I
tff(fact_390_ex__in__conv,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ? [X5: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),A4)
    <=> ( A4 != bot_bo8194388402131092736T_VEBT ) ) ).

% ex_in_conv
tff(fact_391_ex__in__conv,axiom,
    ! [A4: set_real] :
      ( ? [X5: real] : aa_set_real_o(member_real(X5),A4)
    <=> ( A4 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_392_ex__in__conv,axiom,
    ! [A4: set_set_nat] :
      ( ? [X5: set_nat] : aa_set_set_nat_o(member_set_nat(X5),A4)
    <=> ( A4 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_393_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ? [X5: nat] : aa_set_nat_o(member_nat(X5),A4)
    <=> ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_394_ex__in__conv,axiom,
    ! [A4: set_int] :
      ( ? [X5: int] : aa_set_int_o(member_int(X5),A4)
    <=> ( A4 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_395_ex__in__conv,axiom,
    ! [A4: set_o] :
      ( ? [X5: $o] : aa_set_o_o(member_o((X5)),A4)
    <=> ( A4 != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_396_not__psubset__empty,axiom,
    ! [A4: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_397_not__psubset__empty,axiom,
    ! [A4: set_int] : ~ aa_set_int_o(ord_less_set_int(A4),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_398_not__psubset__empty,axiom,
    ! [A4: set_o] : ~ aa_set_o_o(ord_less_set_o(A4),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_399_forall__finite_I1_J,axiom,
    ! [P2: fun_nat_o,I4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),zero_zero_nat)
     => aa_nat_o(P2,I4) ) ).

% forall_finite(1)
tff(fact_400_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Na)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xa)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Tree),Xa)
          | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Tree),Xa) ) ) ) ).

% member_valid_both_member_options
tff(fact_401_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xa)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,T),Xa)
        | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,T),Xa) ) ) ).

% both_member_options_def
tff(fact_402_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xa),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X5) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_403_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xa),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Xa)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_404_order__refl,axiom,
    ! [Xa: set_int] : aa_set_int_o(ord_less_eq_set_int(Xa),Xa) ).

% order_refl
tff(fact_405_order__refl,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Xa) ).

% order_refl
tff(fact_406_order__refl,axiom,
    ! [Xa: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Xa) ).

% order_refl
tff(fact_407_order__refl,axiom,
    ! [Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Xa) ).

% order_refl
tff(fact_408_order__refl,axiom,
    ! [Xa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xa) ).

% order_refl
tff(fact_409_dual__order_Orefl,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% dual_order.refl
tff(fact_410_dual__order_Orefl,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% dual_order.refl
tff(fact_411_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_412_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_413_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_414_buildup__nothing__in__min__max,axiom,
    ! [Na: nat,Xa: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na)),Xa) ).

% buildup_nothing_in_min_max
tff(fact_415_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu2: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf((Uu2),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_416_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv2: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($true,(Uv2))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_417_empty__subsetI,axiom,
    ! [A4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A4) ).

% empty_subsetI
tff(fact_418_empty__subsetI,axiom,
    ! [A4: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A4) ).

% empty_subsetI
tff(fact_419_empty__subsetI,axiom,
    ! [A4: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A4) ).

% empty_subsetI
tff(fact_420_subset__empty,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),bot_bot_set_nat)
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_421_subset__empty,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),bot_bot_set_o)
    <=> ( A4 = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_422_subset__empty,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),bot_bot_set_int)
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_423_psubsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( ( A4 != B4 )
       => aa_set_int_o(ord_less_set_int(A4),B4) ) ) ).

% psubsetI
tff(fact_424_bot__set__def,axiom,
    bot_bot_set_complex = collect_complex(bot_bo8750099748576881071plex_o) ).

% bot_set_def
tff(fact_425_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo8828579575001852285_nat_o) ).

% bot_set_def
tff(fact_426_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_427_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_428_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_429_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_430_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_431_subset__emptyI,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ! [X: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),bot_bo8194388402131092736T_VEBT) ) ).

% subset_emptyI
tff(fact_432_subset__emptyI,axiom,
    ! [A4: set_real] :
      ( ! [X: real] : ~ aa_set_real_o(member_real(X),A4)
     => aa_set_real_o(ord_less_eq_set_real(A4),bot_bot_set_real) ) ).

% subset_emptyI
tff(fact_433_subset__emptyI,axiom,
    ! [A4: set_set_nat] :
      ( ! [X: set_nat] : ~ aa_set_set_nat_o(member_set_nat(X),A4)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),bot_bot_set_set_nat) ) ).

% subset_emptyI
tff(fact_434_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X: nat] : ~ aa_set_nat_o(member_nat(X),A4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_435_subset__emptyI,axiom,
    ! [A4: set_o] :
      ( ! [X: $o] : ~ aa_set_o_o(member_o((X)),A4)
     => aa_set_o_o(ord_less_eq_set_o(A4),bot_bot_set_o) ) ).

% subset_emptyI
tff(fact_436_subset__emptyI,axiom,
    ! [A4: set_int] :
      ( ! [X: int] : ~ aa_set_int_o(member_int(X),A4)
     => aa_set_int_o(ord_less_eq_set_int(A4),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_437_rev__finite__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% rev_finite_subset
tff(fact_438_rev__finite__subset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% rev_finite_subset
tff(fact_439_rev__finite__subset,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ) ).

% rev_finite_subset
tff(fact_440_rev__finite__subset,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => aa_set_int_o(finite_finite_int,A4) ) ) ).

% rev_finite_subset
tff(fact_441_infinite__super,axiom,
    ! [S2: set_nat,T2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T2)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => ~ aa_set_nat_o(finite_finite_nat,T2) ) ) ).

% infinite_super
tff(fact_442_infinite__super,axiom,
    ! [S2: set_complex,T2: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T2) ) ) ).

% infinite_super
tff(fact_443_infinite__super,axiom,
    ! [S2: set_Code_integer,T2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2) ) ) ).

% infinite_super
tff(fact_444_infinite__super,axiom,
    ! [S2: set_int,T2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S2),T2)
     => ( ~ aa_set_int_o(finite_finite_int,S2)
       => ~ aa_set_int_o(finite_finite_int,T2) ) ) ).

% infinite_super
tff(fact_445_finite__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_subset
tff(fact_446_finite__subset,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% finite_subset
tff(fact_447_finite__subset,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ) ).

% finite_subset
tff(fact_448_finite__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => aa_set_int_o(finite_finite_int,A4) ) ) ).

% finite_subset
tff(fact_449_mod__false,axiom,
    ! [H2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(bot_bot_assn),H2) ).

% mod_false
tff(fact_450_subset__iff__psubset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ( aa_set_int_o(ord_less_set_int(A4),B4)
        | ( A4 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_451_subset__psubset__trans,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_set_int(B4),C3)
       => aa_set_int_o(ord_less_set_int(A4),C3) ) ) ).

% subset_psubset_trans
tff(fact_452_subset__not__subset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
        & ~ aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ) ).

% subset_not_subset_eq
tff(fact_453_psubset__subset__trans,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C3)
       => aa_set_int_o(ord_less_set_int(A4),C3) ) ) ).

% psubset_subset_trans
tff(fact_454_psubset__imp__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% psubset_imp_subset
tff(fact_455_psubset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
        & ( A4 != B4 ) ) ) ).

% psubset_eq
tff(fact_456_psubsetE,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
         => aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ) ).

% psubsetE
tff(fact_457_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,Uw2: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_Leaf((Uu2),(Uv2))),Uw2) ).

% VEBT_internal.membermima.simps(1)
tff(fact_458_order__antisym__conv,axiom,
    ! [Y: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),Xa)
     => ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_459_order__antisym__conv,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_460_order__antisym__conv,axiom,
    ! [Y: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_461_order__antisym__conv,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_462_order__antisym__conv,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% order_antisym_conv
tff(fact_463_linorder__le__cases,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% linorder_le_cases
tff(fact_464_linorder__le__cases,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% linorder_le_cases
tff(fact_465_linorder__le__cases,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% linorder_le_cases
tff(fact_466_linorder__le__cases,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% linorder_le_cases
tff(fact_467_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C2 )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_468_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_num(F,B) = C2 )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_469_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C2 )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_470_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(F,B) = C2 )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_471_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_rat(F,B) = C2 )
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_472_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_num(F,B) = C2 )
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_473_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_nat(F,B) = C2 )
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_474_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_int(F,B) = C2 )
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_475_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(F,B) = C2 )
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_476_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_num(F,B) = C2 )
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_477_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_478_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_479_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_480_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_481_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_482_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C2: num] :
      ( ( A = aa_num_num(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_483_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C2: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_484_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C2: num] :
      ( ( A = aa_num_int(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_485_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_486_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C2: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_487_linorder__linear,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% linorder_linear
tff(fact_488_linorder__linear,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% linorder_linear
tff(fact_489_linorder__linear,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% linorder_linear
tff(fact_490_linorder__linear,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% linorder_linear
tff(fact_491_order__eq__refl,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( Xa = Y )
     => aa_set_int_o(ord_less_eq_set_int(Xa),Y) ) ).

% order_eq_refl
tff(fact_492_order__eq__refl,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa = Y )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% order_eq_refl
tff(fact_493_order__eq__refl,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa = Y )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y) ) ).

% order_eq_refl
tff(fact_494_order__eq__refl,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ).

% order_eq_refl
tff(fact_495_order__eq__refl,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ).

% order_eq_refl
tff(fact_496_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_497_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_498_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_499_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_500_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_501_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_502_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_503_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_504_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_505_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C2) ) ) ) ).

% order_subst2
tff(fact_506_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_507_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_508_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_509_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_510_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_511_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_512_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_513_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X)),aa_int_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C2)) ) ) ) ).

% order_subst1
tff(fact_514_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_515_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C2)) ) ) ) ).

% order_subst1
tff(fact_516_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B)
        & aa_set_int_o(ord_less_eq_set_int(B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_517_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_518_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_519_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_520_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_521_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_522_antisym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_523_antisym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_524_antisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_525_antisym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_526_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( aa_set_int_o(ord_less_eq_set_int(C2),B)
       => aa_set_int_o(ord_less_eq_set_int(C2),A) ) ) ).

% dual_order.trans
tff(fact_527_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% dual_order.trans
tff(fact_528_dual__order_Otrans,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% dual_order.trans
tff(fact_529_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% dual_order.trans
tff(fact_530_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% dual_order.trans
tff(fact_531_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( aa_set_int_o(ord_less_eq_set_int(A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_532_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_533_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_534_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_535_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_536_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),A)
        & aa_set_int_o(ord_less_eq_set_int(A),B) ) ) ).

% dual_order.eq_iff
tff(fact_537_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_538_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_539_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_540_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_541_linorder__wlog,axiom,
    ! [P2: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P2,A3),B3) )
     => ( ! [A3: rat,B3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P2,B3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(P2,A3),B3) )
       => aa_rat_o(aa_rat_fun_rat_o(P2,A),B) ) ) ).

% linorder_wlog
tff(fact_542_linorder__wlog,axiom,
    ! [P2: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(P2,A3),B3) )
     => ( ! [A3: num,B3: num] :
            ( aa_num_o(aa_num_fun_num_o(P2,B3),A3)
           => aa_num_o(aa_num_fun_num_o(P2,A3),B3) )
       => aa_num_o(aa_num_fun_num_o(P2,A),B) ) ) ).

% linorder_wlog
tff(fact_543_linorder__wlog,axiom,
    ! [P2: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) )
     => ( ! [A3: nat,B3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P2,B3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) )
       => aa_nat_o(aa_nat_fun_nat_o(P2,A),B) ) ) ).

% linorder_wlog
tff(fact_544_linorder__wlog,axiom,
    ! [P2: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(P2,A3),B3) )
     => ( ! [A3: int,B3: int] :
            ( aa_int_o(aa_int_fun_int_o(P2,B3),A3)
           => aa_int_o(aa_int_fun_int_o(P2,A3),B3) )
       => aa_int_o(aa_int_fun_int_o(P2,A),B) ) ) ).

% linorder_wlog
tff(fact_545_order__trans,axiom,
    ! [Xa: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Y),Z)
       => aa_set_int_o(ord_less_eq_set_int(Xa),Z) ) ) ).

% order_trans
tff(fact_546_order__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z) ) ) ).

% order_trans
tff(fact_547_order__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Z) ) ) ).

% order_trans
tff(fact_548_order__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z) ) ) ).

% order_trans
tff(fact_549_order__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z) ) ) ).

% order_trans
tff(fact_550_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C2)
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% order.trans
tff(fact_551_order_Otrans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% order.trans
tff(fact_552_order_Otrans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% order.trans
tff(fact_553_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% order.trans
tff(fact_554_order_Otrans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% order.trans
tff(fact_555_order__antisym,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_556_order__antisym,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_557_order__antisym,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_558_order__antisym,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_559_order__antisym,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
       => ( Xa = Y ) ) ) ).

% order_antisym
tff(fact_560_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( ( B = C2 )
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_561_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_562_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( B = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_563_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_564_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% ord_le_eq_trans
tff(fact_565_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(ord_less_eq_set_int(B),C2)
       => aa_set_int_o(ord_less_eq_set_int(A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_566_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_567_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_568_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_569_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C2) ) ) ).

% ord_eq_le_trans
tff(fact_570_order__class_Oorder__eq__iff,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( Xa = Y )
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
        & aa_set_int_o(ord_less_eq_set_int(Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_571_order__class_Oorder__eq__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa = Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_572_order__class_Oorder__eq__iff,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa = Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_573_order__class_Oorder__eq__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_574_order__class_Oorder__eq__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ) ).

% order_class.order_eq_iff
tff(fact_575_le__cases3,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xa) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xa)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_576_le__cases3,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xa) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xa)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_577_le__cases3,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xa) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xa)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_578_le__cases3,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xa) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xa)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_579_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_580_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_581_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_582_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_583_order__less__imp__not__less,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_584_order__less__imp__not__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_585_order__less__imp__not__less,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_586_order__less__imp__not__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_587_order__less__imp__not__less,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% order_less_imp_not_less
tff(fact_588_order__less__imp__not__eq2,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_589_order__less__imp__not__eq2,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_590_order__less__imp__not__eq2,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_591_order__less__imp__not__eq2,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_592_order__less__imp__not__eq2,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( Y != Xa ) ) ).

% order_less_imp_not_eq2
tff(fact_593_order__less__imp__not__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_594_order__less__imp__not__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_595_order__less__imp__not__eq,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_596_order__less__imp__not__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_597_order__less__imp__not__eq,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( Xa != Y ) ) ).

% order_less_imp_not_eq
tff(fact_598_linorder__less__linear,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
      | ( Xa = Y )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% linorder_less_linear
tff(fact_599_linorder__less__linear,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
      | ( Xa = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% linorder_less_linear
tff(fact_600_linorder__less__linear,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
      | ( Xa = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% linorder_less_linear
tff(fact_601_linorder__less__linear,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
      | ( Xa = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% linorder_less_linear
tff(fact_602_linorder__less__linear,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
      | ( Xa = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% linorder_less_linear
tff(fact_603_order__less__imp__triv,axiom,
    ! [Xa: real,Y: real,P2: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_604_order__less__imp__triv,axiom,
    ! [Xa: rat,Y: rat,P2: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_605_order__less__imp__triv,axiom,
    ! [Xa: num,Y: num,P2: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_606_order__less__imp__triv,axiom,
    ! [Xa: nat,Y: nat,P2: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_607_order__less__imp__triv,axiom,
    ! [Xa: int,Y: int,P2: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa)
       => (P2) ) ) ).

% order_less_imp_triv
tff(fact_608_order__less__not__sym,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% order_less_not_sym
tff(fact_609_order__less__not__sym,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% order_less_not_sym
tff(fact_610_order__less__not__sym,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% order_less_not_sym
tff(fact_611_order__less__not__sym,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% order_less_not_sym
tff(fact_612_order__less__not__sym,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% order_less_not_sym
tff(fact_613_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_614_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_615_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_616_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_617_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_618_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_619_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_620_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_621_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_622_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C2) ) ) ) ).

% order_less_subst2
tff(fact_623_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_624_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_625_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_626_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_627_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_628_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_629_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_630_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_631_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_632_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C2)) ) ) ) ).

% order_less_subst1
tff(fact_633_order__less__irrefl,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Xa) ).

% order_less_irrefl
tff(fact_634_order__less__irrefl,axiom,
    ! [Xa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Xa) ).

% order_less_irrefl
tff(fact_635_order__less__irrefl,axiom,
    ! [Xa: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Xa) ).

% order_less_irrefl
tff(fact_636_order__less__irrefl,axiom,
    ! [Xa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Xa) ).

% order_less_irrefl
tff(fact_637_order__less__irrefl,axiom,
    ! [Xa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xa) ).

% order_less_irrefl
tff(fact_638_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,B) = C2 )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_639_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_rat(F,B) = C2 )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_640_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_num(F,B) = C2 )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_641_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_nat(F,B) = C2 )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_642_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_int(F,B) = C2 )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_643_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_real(F,B) = C2 )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_644_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C2 )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_645_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(F,B) = C2 )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_646_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C2 )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_647_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_int(F,B) = C2 )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_648_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C2: real] :
      ( ( A = aa_real_real(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_649_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C2: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_650_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C2: real] :
      ( ( A = aa_real_num(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_651_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C2: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_652_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C2: real] :
      ( ( A = aa_real_int(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_653_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_654_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_655_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_656_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_657_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C2: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_658_order__less__trans,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z) ) ) ).

% order_less_trans
tff(fact_659_order__less__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z) ) ) ).

% order_less_trans
tff(fact_660_order__less__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z) ) ) ).

% order_less_trans
tff(fact_661_order__less__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z) ) ) ).

% order_less_trans
tff(fact_662_order__less__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z) ) ) ).

% order_less_trans
tff(fact_663_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order_less_asym'
tff(fact_664_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order_less_asym'
tff(fact_665_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_666_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order_less_asym'
tff(fact_667_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order_less_asym'
tff(fact_668_linorder__neq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_669_linorder__neq__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_670_linorder__neq__iff,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_671_linorder__neq__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_672_linorder__neq__iff,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_neq_iff
tff(fact_673_order__less__asym,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% order_less_asym
tff(fact_674_order__less__asym,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% order_less_asym
tff(fact_675_order__less__asym,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% order_less_asym
tff(fact_676_order__less__asym,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% order_less_asym
tff(fact_677_order__less__asym,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% order_less_asym
tff(fact_678_linorder__neqE,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_679_linorder__neqE,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_680_linorder__neqE,axiom,
    ! [Xa: num,Y: num] :
      ( ( Xa != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_681_linorder__neqE,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_682_linorder__neqE,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_neqE
tff(fact_683_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_684_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_685_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_686_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_687_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_688_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_689_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_690_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_691_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_692_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_693_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_694_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_695_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_696_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_697_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans
tff(fact_698_not__less__iff__gr__or__eq,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_699_not__less__iff__gr__or__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_700_not__less__iff__gr__or__eq,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_701_not__less__iff__gr__or__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_702_not__less__iff__gr__or__eq,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa)
        | ( Xa = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_703_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% order.strict_trans
tff(fact_704_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% order.strict_trans
tff(fact_705_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% order.strict_trans
tff(fact_706_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans
tff(fact_707_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans
tff(fact_708_linorder__less__wlog,axiom,
    ! [P2: fun_real_fun_real_o,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B3)
         => aa_real_o(aa_real_fun_real_o(P2,A3),B3) )
     => ( ! [A3: real] : aa_real_o(aa_real_fun_real_o(P2,A3),A3)
       => ( ! [A3: real,B3: real] :
              ( aa_real_o(aa_real_fun_real_o(P2,B3),A3)
             => aa_real_o(aa_real_fun_real_o(P2,A3),B3) )
         => aa_real_o(aa_real_fun_real_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_709_linorder__less__wlog,axiom,
    ! [P2: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P2,A3),B3) )
     => ( ! [A3: rat] : aa_rat_o(aa_rat_fun_rat_o(P2,A3),A3)
       => ( ! [A3: rat,B3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P2,B3),A3)
             => aa_rat_o(aa_rat_fun_rat_o(P2,A3),B3) )
         => aa_rat_o(aa_rat_fun_rat_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_710_linorder__less__wlog,axiom,
    ! [P2: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(P2,A3),B3) )
     => ( ! [A3: num] : aa_num_o(aa_num_fun_num_o(P2,A3),A3)
       => ( ! [A3: num,B3: num] :
              ( aa_num_o(aa_num_fun_num_o(P2,B3),A3)
             => aa_num_o(aa_num_fun_num_o(P2,A3),B3) )
         => aa_num_o(aa_num_fun_num_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_711_linorder__less__wlog,axiom,
    ! [P2: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,A3),A3)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P2,B3),A3)
             => aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_712_linorder__less__wlog,axiom,
    ! [P2: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(P2,A3),B3) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(P2,A3),A3)
       => ( ! [A3: int,B3: int] :
              ( aa_int_o(aa_int_fun_int_o(P2,B3),A3)
             => aa_int_o(aa_int_fun_int_o(P2,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(P2,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_713_exists__least__iff,axiom,
    ! [P2: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P2,X_12)
    <=> ? [N5: nat] :
          ( aa_nat_o(P2,N5)
          & ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N5)
             => ~ aa_nat_o(P2,M5) ) ) ) ).

% exists_least_iff
tff(fact_714_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_715_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_716_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_717_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_718_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_719_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% dual_order.asym
tff(fact_720_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% dual_order.asym
tff(fact_721_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_722_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_723_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% dual_order.asym
tff(fact_724_linorder__cases,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( ( Xa != Y )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_cases
tff(fact_725_linorder__cases,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( ( Xa != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_cases
tff(fact_726_linorder__cases,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( ( Xa != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ) ).

% linorder_cases
tff(fact_727_linorder__cases,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( ( Xa != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ) ).

% linorder_cases
tff(fact_728_linorder__cases,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( ( Xa != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_cases
tff(fact_729_antisym__conv3,axiom,
    ! [Y: real,Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_730_antisym__conv3,axiom,
    ! [Y: rat,Xa: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_731_antisym__conv3,axiom,
    ! [Y: num,Xa: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_732_antisym__conv3,axiom,
    ! [Y: nat,Xa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_733_antisym__conv3,axiom,
    ! [Y: int,Xa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv3
tff(fact_734_less__induct,axiom,
    ! [P2: fun_nat_o,A: nat] :
      ( ! [X: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X)
             => aa_nat_o(P2,Y4) )
         => aa_nat_o(P2,X) )
     => aa_nat_o(P2,A) ) ).

% less_induct
tff(fact_735_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( B = C2 )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_736_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_737_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_738_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_739_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% ord_less_eq_trans
tff(fact_740_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_741_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_742_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_743_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_744_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% ord_eq_less_trans
tff(fact_745_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order.asym
tff(fact_746_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order.asym
tff(fact_747_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_748_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order.asym
tff(fact_749_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order.asym
tff(fact_750_less__imp__neq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_751_less__imp__neq,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_752_less__imp__neq,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_753_less__imp__neq,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_754_less__imp__neq,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( Xa != Y ) ) ).

% less_imp_neq
tff(fact_755_dense,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Y) ) ) ).

% dense
tff(fact_756_dense,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ? [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z3)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),Y) ) ) ).

% dense
tff(fact_757_gt__ex,axiom,
    ! [Xa: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X_1) ).

% gt_ex
tff(fact_758_gt__ex,axiom,
    ! [Xa: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X_1) ).

% gt_ex
tff(fact_759_gt__ex,axiom,
    ! [Xa: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X_1) ).

% gt_ex
tff(fact_760_gt__ex,axiom,
    ! [Xa: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X_1) ).

% gt_ex
tff(fact_761_lt__ex,axiom,
    ! [Xa: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xa) ).

% lt_ex
tff(fact_762_lt__ex,axiom,
    ! [Xa: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xa) ).

% lt_ex
tff(fact_763_lt__ex,axiom,
    ! [Xa: int] :
    ? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xa) ).

% lt_ex
tff(fact_764_order__le__imp__less__or__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_765_order__le__imp__less__or__eq,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
     => ( aa_set_int_o(ord_less_set_int(Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_766_order__le__imp__less__or__eq,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_767_order__le__imp__less__or__eq,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_768_order__le__imp__less__or__eq,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_769_order__le__imp__less__or__eq,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_770_linorder__le__less__linear,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_771_linorder__le__less__linear,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_772_linorder__le__less__linear,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_773_linorder__le__less__linear,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_774_linorder__le__less__linear,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% linorder_le_less_linear
tff(fact_775_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_776_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_777_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C2: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_778_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_779_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C2: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C2)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_780_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B)),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_781_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_782_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_783_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_784_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C2)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_785_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_786_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_787_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_788_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_789_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_790_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_791_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_792_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_793_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_794_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_795_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_796_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_797_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_798_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_799_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_800_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C2: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_801_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_802_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_803_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_804_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B)),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_805_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_806_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_807_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_808_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_809_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_810_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_811_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_812_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_813_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_814_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_815_order__less__le__trans,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_816_order__less__le__trans,axiom,
    ! [Xa: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Y),Z)
       => aa_set_int_o(ord_less_set_int(Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_817_order__less__le__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_818_order__less__le__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_819_order__less__le__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_820_order__less__le__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z) ) ) ).

% order_less_le_trans
tff(fact_821_order__le__less__trans,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_822_order__le__less__trans,axiom,
    ! [Xa: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
     => ( aa_set_int_o(ord_less_set_int(Y),Z)
       => aa_set_int_o(ord_less_set_int(Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_823_order__le__less__trans,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_824_order__le__less__trans,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_825_order__le__less__trans,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_826_order__le__less__trans,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z) ) ) ).

% order_le_less_trans
tff(fact_827_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_neq_le_trans
tff(fact_828_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( aa_set_int_o(ord_less_eq_set_int(A),B)
       => aa_set_int_o(ord_less_set_int(A),B) ) ) ).

% order_neq_le_trans
tff(fact_829_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_830_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_831_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_832_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_833_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( A != B )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_le_neq_trans
tff(fact_834_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( ( A != B )
       => aa_set_int_o(ord_less_set_int(A),B) ) ) ).

% order_le_neq_trans
tff(fact_835_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( A != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_836_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_837_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_838_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_839_order__less__imp__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% order_less_imp_le
tff(fact_840_order__less__imp__le,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),Y)
     => aa_set_int_o(ord_less_eq_set_int(Xa),Y) ) ).

% order_less_imp_le
tff(fact_841_order__less__imp__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% order_less_imp_le
tff(fact_842_order__less__imp__le,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y) ) ).

% order_less_imp_le
tff(fact_843_order__less__imp__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ).

% order_less_imp_le
tff(fact_844_order__less__imp__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ).

% order_less_imp_le
tff(fact_845_linorder__not__less,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ).

% linorder_not_less
tff(fact_846_linorder__not__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% linorder_not_less
tff(fact_847_linorder__not__less,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% linorder_not_less
tff(fact_848_linorder__not__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% linorder_not_less
tff(fact_849_linorder__not__less,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% linorder_not_less
tff(fact_850_linorder__not__le,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ).

% linorder_not_le
tff(fact_851_linorder__not__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ).

% linorder_not_le
tff(fact_852_linorder__not__le,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xa) ) ).

% linorder_not_le
tff(fact_853_linorder__not__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa) ) ).

% linorder_not_le
tff(fact_854_linorder__not__le,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ).

% linorder_not_le
tff(fact_855_order__less__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_856_order__less__le,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),Y)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_857_order__less__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_858_order__less__le,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_859_order__less__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_860_order__less__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & ( Xa != Y ) ) ) ).

% order_less_le
tff(fact_861_order__le__less,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_862_order__le__less,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
    <=> ( aa_set_int_o(ord_less_set_int(Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_863_order__le__less,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_864_order__le__less,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_865_order__le__less,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_866_order__le__less,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
        | ( Xa = Y ) ) ) ).

% order_le_less
tff(fact_867_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_868_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
     => aa_set_int_o(ord_less_eq_set_int(B),A) ) ).

% dual_order.strict_implies_order
tff(fact_869_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_870_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_871_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_872_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_873_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% order.strict_implies_order
tff(fact_874_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
     => aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% order.strict_implies_order
tff(fact_875_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% order.strict_implies_order
tff(fact_876_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_877_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_878_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% order.strict_implies_order
tff(fact_879_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_880_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),A)
        & ~ aa_set_int_o(ord_less_eq_set_int(A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_881_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_882_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_883_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_884_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_885_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_886_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
     => ( aa_set_int_o(ord_less_eq_set_int(C2),B)
       => aa_set_int_o(ord_less_set_int(C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_887_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_888_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_889_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_890_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans2
tff(fact_891_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_892_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( aa_set_int_o(ord_less_set_int(C2),B)
       => aa_set_int_o(ord_less_set_int(C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_893_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_894_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_895_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_896_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% dual_order.strict_trans1
tff(fact_897_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_898_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_set_int(B),A)
    <=> ( aa_set_int_o(ord_less_eq_set_int(B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_899_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_900_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_901_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_902_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_903_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_904_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
    <=> ( aa_set_int_o(ord_less_set_int(B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_905_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_906_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_907_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_908_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_909_dense__le__bounded,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_910_dense__le__bounded,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_911_dense__ge__bounded,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xa)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Xa)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),W) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_912_dense__ge__bounded,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xa)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Xa)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_913_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% order.strict_iff_not
tff(fact_914_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B)
        & ~ aa_set_int_o(ord_less_eq_set_int(B),A) ) ) ).

% order.strict_iff_not
tff(fact_915_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_916_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_917_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_918_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_919_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% order.strict_trans2
tff(fact_920_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
     => ( aa_set_int_o(ord_less_eq_set_int(B),C2)
       => aa_set_int_o(ord_less_set_int(A),C2) ) ) ).

% order.strict_trans2
tff(fact_921_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% order.strict_trans2
tff(fact_922_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% order.strict_trans2
tff(fact_923_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans2
tff(fact_924_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans2
tff(fact_925_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2) ) ) ).

% order.strict_trans1
tff(fact_926_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(ord_less_set_int(B),C2)
       => aa_set_int_o(ord_less_set_int(A),C2) ) ) ).

% order.strict_trans1
tff(fact_927_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C2) ) ) ).

% order.strict_trans1
tff(fact_928_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C2) ) ) ).

% order.strict_trans1
tff(fact_929_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C2) ) ) ).

% order.strict_trans1
tff(fact_930_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C2) ) ) ).

% order.strict_trans1
tff(fact_931_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_932_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_933_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_934_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_935_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_936_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_937_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_938_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
    <=> ( aa_set_int_o(ord_less_set_int(A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_939_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_940_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_941_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_942_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_943_not__le__imp__less,axiom,
    ! [Y: real,Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% not_le_imp_less
tff(fact_944_not__le__imp__less,axiom,
    ! [Y: rat,Xa: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ).

% not_le_imp_less
tff(fact_945_not__le__imp__less,axiom,
    ! [Y: num,Xa: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y) ) ).

% not_le_imp_less
tff(fact_946_not__le__imp__less,axiom,
    ! [Y: nat,Xa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ).

% not_le_imp_less
tff(fact_947_not__le__imp__less,axiom,
    ! [Y: int,Xa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ).

% not_le_imp_less
tff(fact_948_less__le__not__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_949_less__le__not__le,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xa),Y)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
        & ~ aa_set_int_o(ord_less_eq_set_int(Y),Xa) ) ) ).

% less_le_not_le
tff(fact_950_less__le__not__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_951_less__le__not__le,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_952_less__le__not__le,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_953_less__le__not__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ) ).

% less_le_not_le
tff(fact_954_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_le
tff(fact_955_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_956_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_ge
tff(fact_957_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_ge
tff(fact_958_antisym__conv2,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_959_antisym__conv2,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
     => ( ~ aa_set_int_o(ord_less_set_int(Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_960_antisym__conv2,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_961_antisym__conv2,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_962_antisym__conv2,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_963_antisym__conv2,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv2
tff(fact_964_antisym__conv1,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_965_antisym__conv1,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(Xa),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_966_antisym__conv1,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_967_antisym__conv1,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_968_antisym__conv1,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_969_antisym__conv1,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
      <=> ( Xa = Y ) ) ) ).

% antisym_conv1
tff(fact_970_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_971_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(A),B)
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_972_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_973_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_974_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_975_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_976_leI,axiom,
    ! [Xa: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ).

% leI
tff(fact_977_leI,axiom,
    ! [Xa: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa) ) ).

% leI
tff(fact_978_leI,axiom,
    ! [Xa: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa) ) ).

% leI
tff(fact_979_leI,axiom,
    ! [Xa: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa) ) ).

% leI
tff(fact_980_leI,axiom,
    ! [Xa: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa) ) ).

% leI
tff(fact_981_leD,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% leD
tff(fact_982_leD,axiom,
    ! [Y: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),Xa)
     => ~ aa_set_int_o(ord_less_set_int(Xa),Y) ) ).

% leD
tff(fact_983_leD,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ).

% leD
tff(fact_984_leD,axiom,
    ! [Y: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y) ) ).

% leD
tff(fact_985_leD,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ).

% leD
tff(fact_986_leD,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ).

% leD
tff(fact_987_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_988_bot_Oextremum__uniqueI,axiom,
    ! [A: assn] :
      ( ord_less_eq_assn(A,bot_bot_assn)
     => ( A = bot_bot_assn ) ) ).

% bot.extremum_uniqueI
tff(fact_989_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_990_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_991_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_992_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_993_bot_Oextremum__unique,axiom,
    ! [A: assn] :
      ( ord_less_eq_assn(A,bot_bot_assn)
    <=> ( A = bot_bot_assn ) ) ).

% bot.extremum_unique
tff(fact_994_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_995_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_996_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_997_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_998_bot_Oextremum,axiom,
    ! [A: assn] : ord_less_eq_assn(bot_bot_assn,A) ).

% bot.extremum
tff(fact_999_bot_Oextremum,axiom,
    ! [A: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A) ).

% bot.extremum
tff(fact_1000_bot_Oextremum,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% bot.extremum
tff(fact_1001_bot_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).

% bot.extremum
tff(fact_1002_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1003_bot_Onot__eq__extremum,axiom,
    ! [A: assn] :
      ( ( A != bot_bot_assn )
    <=> ord_less_assn(bot_bot_assn,A) ) ).

% bot.not_eq_extremum
tff(fact_1004_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),A) ) ).

% bot.not_eq_extremum
tff(fact_1005_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
    <=> aa_set_o_o(ord_less_set_o(bot_bot_set_o),A) ) ).

% bot.not_eq_extremum
tff(fact_1006_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1007_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_1008_bot_Oextremum__strict,axiom,
    ! [A: assn] : ~ ord_less_assn(A,bot_bot_assn) ).

% bot.extremum_strict
tff(fact_1009_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ aa_set_int_o(ord_less_set_int(A),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_1010_bot_Oextremum__strict,axiom,
    ! [A: set_o] : ~ aa_set_o_o(ord_less_set_o(A),bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_1011_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_1012_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1013_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),one_one_nat)
    <=> ? [A6: $o,B6: $o] : ( T = vEBT_Leaf((A6),(B6)) ) ) ).

% deg1Leaf
tff(fact_1014_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),one_one_nat)
     => ? [A3: $o,B3: $o] : ( T = vEBT_Leaf((A3),(B3)) ) ) ).

% deg_1_Leaf
tff(fact_1015_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( Na = one_one_nat )
       => ? [A3: $o,B3: $o] : ( T = vEBT_Leaf((A3),(B3)) ) ) ) ).

% deg_1_Leafy
tff(fact_1016_finite__nat__set__iff__bounded__le,axiom,
    ! [N6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N6)
    <=> ? [M5: nat] :
        ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),N6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1017_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M5: nat] :
        ? [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N5)
          & aa_set_nat_o(member_nat(N5),S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_1018_finite__nat__set__iff__bounded,axiom,
    ! [N6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N6)
    <=> ? [M5: nat] :
        ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),N6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1019_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M5: nat] :
        ? [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N5)
          & aa_set_nat_o(member_nat(N5),S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_1020_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,Na: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Na) )
     => aa_set_nat_o(finite_finite_nat,N6) ) ).

% bounded_nat_set_is_finite
tff(fact_1021_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M3)
         => ? [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N7)
              & aa_set_nat_o(member_nat(N7),S2) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S2) ) ).

% unbounded_k_infinite
tff(fact_1022_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] : ( Xa != vEBT_Leaf((A3),(B3)) )
     => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg2,TreeList2),Summary2) ) ) ).

% VEBT_internal.height.cases
tff(fact_1023_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(F,lattic8794016678065449205x_real(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1024_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ~ ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Code_integer_real(F,X3)),aa_Code_integer_real(F,lattic2659822949269061924r_real(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1025_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1026_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X3: int] :
              ( aa_set_int_o(member_int(X3),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,lattic2675449441010098035t_real(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1027_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ( S2 != bot_bot_set_o )
       => ~ ? [X3: $o] :
              ( aa_set_o_o(member_o((X3)),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(F,(X3))),aa_o_real(F,aa_set_o_o(lattic8697145971487455083o_real(F),S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1028_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1029_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ~ ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(F,X3)),aa_Code_integer_rat(F,lattic5439806495466278992er_rat(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1030_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1031_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X3: int] :
              ( aa_set_int_o(member_int(X3),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1032_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ( S2 != bot_bot_set_o )
       => ~ ? [X3: $o] :
              ( aa_set_o_o(member_o((X3)),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_o_rat(F,(X3))),aa_o_rat(F,aa_set_o_o(lattic2140725968369957399_o_rat(F),S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1033_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_nat_o(member_nat(X),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ).

% subsetI
tff(fact_1034_subsetI,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B4) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4) ) ).

% subsetI
tff(fact_1035_subsetI,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_set_real_o(member_real(X),B4) )
     => aa_set_real_o(ord_less_eq_set_real(A4),B4) ) ).

% subsetI
tff(fact_1036_subsetI,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),A4)
         => aa_set_set_nat_o(member_set_nat(X),B4) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4) ) ).

% subsetI
tff(fact_1037_subsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_int_o(member_int(X),B4) )
     => aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% subsetI
tff(fact_1038_subset__antisym,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( A4 = B4 ) ) ) ).

% subset_antisym
tff(fact_1039_less__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),one_one_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% less_one
tff(fact_1040_subset__Collect__conv,axiom,
    ! [S2: set_complex,P2: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S2),collect_complex(P2))
    <=> ! [X5: complex] :
          ( aa_set_complex_o(member_complex(X5),S2)
         => aa_complex_o(P2,X5) ) ) ).

% subset_Collect_conv
tff(fact_1041_subset__Collect__conv,axiom,
    ! [S2: set_list_nat,P2: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(S2,collect_list_nat(P2))
    <=> ! [X5: list_nat] :
          ( member_list_nat(X5,S2)
         => aa_list_nat_o(P2,X5) ) ) ).

% subset_Collect_conv
tff(fact_1042_subset__Collect__conv,axiom,
    ! [S2: set_set_nat,P2: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(S2),collect_set_nat(P2))
    <=> ! [X5: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X5),S2)
         => aa_set_nat_o(P2,X5) ) ) ).

% subset_Collect_conv
tff(fact_1043_subset__Collect__conv,axiom,
    ! [S2: set_nat,P2: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),collect_nat(P2))
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),S2)
         => aa_nat_o(P2,X5) ) ) ).

% subset_Collect_conv
tff(fact_1044_subset__Collect__conv,axiom,
    ! [S2: set_int,P2: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(S2),collect_int(P2))
    <=> ! [X5: int] :
          ( aa_set_int_o(member_int(X5),S2)
         => aa_int_o(P2,X5) ) ) ).

% subset_Collect_conv
tff(fact_1045_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,Xa: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_nat_o(member_nat(Xa),B4) ) ) ).

% in_mono
tff(fact_1046_in__mono,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B4) ) ) ).

% in_mono
tff(fact_1047_in__mono,axiom,
    ! [A4: set_real,B4: set_real,Xa: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => aa_set_real_o(member_real(Xa),B4) ) ) ).

% in_mono
tff(fact_1048_in__mono,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A4)
       => aa_set_set_nat_o(member_set_nat(Xa),B4) ) ) ).

% in_mono
tff(fact_1049_in__mono,axiom,
    ! [A4: set_int,B4: set_int,Xa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_set_int_o(member_int(Xa),B4) ) ) ).

% in_mono
tff(fact_1050_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(member_nat(C2),A4)
       => aa_set_nat_o(member_nat(C2),B4) ) ) ).

% subsetD
tff(fact_1051_subsetD,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A4)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B4) ) ) ).

% subsetD
tff(fact_1052_subsetD,axiom,
    ! [A4: set_real,B4: set_real,C2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
     => ( aa_set_real_o(member_real(C2),A4)
       => aa_set_real_o(member_real(C2),B4) ) ) ).

% subsetD
tff(fact_1053_subsetD,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,C2: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
     => ( aa_set_set_nat_o(member_set_nat(C2),A4)
       => aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% subsetD
tff(fact_1054_subsetD,axiom,
    ! [A4: set_int,B4: set_int,C2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(member_int(C2),A4)
       => aa_set_int_o(member_int(C2),B4) ) ) ).

% subsetD
tff(fact_1055_equalityE,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
         => ~ aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ) ).

% equalityE
tff(fact_1056_subset__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_set_nat_o(member_nat(X5),B4) ) ) ).

% subset_eq
tff(fact_1057_subset__eq,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
    <=> ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),A4)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),B4) ) ) ).

% subset_eq
tff(fact_1058_subset__eq,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
    <=> ! [X5: real] :
          ( aa_set_real_o(member_real(X5),A4)
         => aa_set_real_o(member_real(X5),B4) ) ) ).

% subset_eq
tff(fact_1059_subset__eq,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
    <=> ! [X5: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X5),A4)
         => aa_set_set_nat_o(member_set_nat(X5),B4) ) ) ).

% subset_eq
tff(fact_1060_subset__eq,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_set_int_o(member_int(X5),B4) ) ) ).

% subset_eq
tff(fact_1061_equalityD1,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% equalityD1
tff(fact_1062_Set_OequalityD2,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
     => aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ).

% Set.equalityD2
tff(fact_1063_subset__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ! [T3: nat] :
          ( aa_set_nat_o(member_nat(T3),A4)
         => aa_set_nat_o(member_nat(T3),B4) ) ) ).

% subset_iff
tff(fact_1064_subset__iff,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
    <=> ! [T3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T3),A4)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T3),B4) ) ) ).

% subset_iff
tff(fact_1065_subset__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
    <=> ! [T3: real] :
          ( aa_set_real_o(member_real(T3),A4)
         => aa_set_real_o(member_real(T3),B4) ) ) ).

% subset_iff
tff(fact_1066_subset__iff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
    <=> ! [T3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(T3),A4)
         => aa_set_set_nat_o(member_set_nat(T3),B4) ) ) ).

% subset_iff
tff(fact_1067_subset__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ! [T3: int] :
          ( aa_set_int_o(member_int(T3),A4)
         => aa_set_int_o(member_int(T3),B4) ) ) ).

% subset_iff
tff(fact_1068_subset__refl,axiom,
    ! [A4: set_int] : aa_set_int_o(ord_less_eq_set_int(A4),A4) ).

% subset_refl
tff(fact_1069_Collect__mono,axiom,
    ! [P2: fun_complex_o,Q2: fun_complex_o] :
      ( ! [X: complex] :
          ( aa_complex_o(P2,X)
         => aa_complex_o(Q2,X) )
     => aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P2)),collect_complex(Q2)) ) ).

% Collect_mono
tff(fact_1070_Collect__mono,axiom,
    ! [P2: fun_list_nat_o,Q2: fun_list_nat_o] :
      ( ! [X: list_nat] :
          ( aa_list_nat_o(P2,X)
         => aa_list_nat_o(Q2,X) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P2),collect_list_nat(Q2)) ) ).

% Collect_mono
tff(fact_1071_Collect__mono,axiom,
    ! [P2: fun_set_nat_o,Q2: fun_set_nat_o] :
      ( ! [X: set_nat] :
          ( aa_set_nat_o(P2,X)
         => aa_set_nat_o(Q2,X) )
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P2)),collect_set_nat(Q2)) ) ).

% Collect_mono
tff(fact_1072_Collect__mono,axiom,
    ! [P2: fun_nat_o,Q2: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
         => aa_nat_o(Q2,X) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P2)),collect_nat(Q2)) ) ).

% Collect_mono
tff(fact_1073_Collect__mono,axiom,
    ! [P2: fun_int_o,Q2: fun_int_o] :
      ( ! [X: int] :
          ( aa_int_o(P2,X)
         => aa_int_o(Q2,X) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P2)),collect_int(Q2)) ) ).

% Collect_mono
tff(fact_1074_subset__trans,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C3)
       => aa_set_int_o(ord_less_eq_set_int(A4),C3) ) ) ).

% subset_trans
tff(fact_1075_set__eq__subset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( A4 = B4 )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
        & aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ) ).

% set_eq_subset
tff(fact_1076_Collect__mono__iff,axiom,
    ! [P2: fun_complex_o,Q2: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P2)),collect_complex(Q2))
    <=> ! [X5: complex] :
          ( aa_complex_o(P2,X5)
         => aa_complex_o(Q2,X5) ) ) ).

% Collect_mono_iff
tff(fact_1077_Collect__mono__iff,axiom,
    ! [P2: fun_list_nat_o,Q2: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P2),collect_list_nat(Q2))
    <=> ! [X5: list_nat] :
          ( aa_list_nat_o(P2,X5)
         => aa_list_nat_o(Q2,X5) ) ) ).

% Collect_mono_iff
tff(fact_1078_Collect__mono__iff,axiom,
    ! [P2: fun_set_nat_o,Q2: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(P2)),collect_set_nat(Q2))
    <=> ! [X5: set_nat] :
          ( aa_set_nat_o(P2,X5)
         => aa_set_nat_o(Q2,X5) ) ) ).

% Collect_mono_iff
tff(fact_1079_Collect__mono__iff,axiom,
    ! [P2: fun_nat_o,Q2: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P2)),collect_nat(Q2))
    <=> ! [X5: nat] :
          ( aa_nat_o(P2,X5)
         => aa_nat_o(Q2,X5) ) ) ).

% Collect_mono_iff
tff(fact_1080_Collect__mono__iff,axiom,
    ! [P2: fun_int_o,Q2: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(P2)),collect_int(Q2))
    <=> ! [X5: int] :
          ( aa_int_o(P2,X5)
         => aa_int_o(Q2,X5) ) ) ).

% Collect_mono_iff
tff(fact_1081_one__reorient,axiom,
    ! [Xa: uint32] :
      ( ( one_one_uint32 = Xa )
    <=> ( Xa = one_one_uint32 ) ) ).

% one_reorient
tff(fact_1082_one__reorient,axiom,
    ! [Xa: real] :
      ( ( one_one_real = Xa )
    <=> ( Xa = one_one_real ) ) ).

% one_reorient
tff(fact_1083_one__reorient,axiom,
    ! [Xa: rat] :
      ( ( one_one_rat = Xa )
    <=> ( Xa = one_one_rat ) ) ).

% one_reorient
tff(fact_1084_one__reorient,axiom,
    ! [Xa: nat] :
      ( ( one_one_nat = Xa )
    <=> ( Xa = one_one_nat ) ) ).

% one_reorient
tff(fact_1085_one__reorient,axiom,
    ! [Xa: int] :
      ( ( one_one_int = Xa )
    <=> ( Xa = one_one_int ) ) ).

% one_reorient
tff(fact_1086_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_1087_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1088_nat__geq__1__eq__neqz,axiom,
    ! [Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Xa)
    <=> ( Xa != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_1089_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,vEBT_Leaf((Uu2),(Uv2))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_1090_nat__induct__non__zero,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(P2,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P2,N)
               => aa_nat_o(P2,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P2,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_1091_bounded__Max__nat,axiom,
    ! [P2: fun_nat_o,Xa: nat,M7: nat] :
      ( aa_nat_o(P2,Xa)
     => ( ! [X: nat] :
            ( aa_nat_o(P2,X)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M7) )
       => ~ ! [M3: nat] :
              ( aa_nat_o(P2,M3)
             => ~ ! [X3: nat] :
                    ( aa_nat_o(P2,X3)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M3) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1092_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_Leaf((A),(B))),Xa)
    <=> $ite(
          Xa = zero_zero_nat,
          (A),
          $ite(Xa = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_1093_finite__transitivity__chain,axiom,
    ! [A4: set_VEBT_VEBT,R2: fun_VE2962686792283380846VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( ! [X: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R2,X),X)
       => ( ! [X: vEBT_VEBT,Y3: vEBT_VEBT,Z3: vEBT_VEBT] :
              ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R2,X),Y3)
             => ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R2,Y3),Z3)
               => aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R2,X),Z3) ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
               => ? [Y4: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),A4)
                    & aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R2,X),Y4) ) )
           => ( A4 = bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1094_finite__transitivity__chain,axiom,
    ! [A4: set_real,R2: fun_real_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [X: real] : ~ aa_real_o(aa_real_fun_real_o(R2,X),X)
       => ( ! [X: real,Y3: real,Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(R2,X),Y3)
             => ( aa_real_o(aa_real_fun_real_o(R2,Y3),Z3)
               => aa_real_o(aa_real_fun_real_o(R2,X),Z3) ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),A4)
               => ? [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),A4)
                    & aa_real_o(aa_real_fun_real_o(R2,X),Y4) ) )
           => ( A4 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1095_finite__transitivity__chain,axiom,
    ! [A4: set_set_nat,R2: fun_se2170385732985038782_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( ! [X: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(R2,X),X)
       => ( ! [X: set_nat,Y3: set_nat,Z3: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R2,X),Y3)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R2,Y3),Z3)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(R2,X),Z3) ) )
         => ( ! [X: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(X),A4)
               => ? [Y4: set_nat] :
                    ( aa_set_set_nat_o(member_set_nat(Y4),A4)
                    & aa_set_nat_o(aa_set7861427066675100071_nat_o(R2,X),Y4) ) )
           => ( A4 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1096_finite__transitivity__chain,axiom,
    ! [A4: set_complex,R2: fun_co1148497455491383662plex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X: complex] : ~ aa_complex_o(aa_com5808801942499186519plex_o(R2,X),X)
       => ( ! [X: complex,Y3: complex,Z3: complex] :
              ( aa_complex_o(aa_com5808801942499186519plex_o(R2,X),Y3)
             => ( aa_complex_o(aa_com5808801942499186519plex_o(R2,Y3),Z3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R2,X),Z3) ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),A4)
               => ? [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),A4)
                    & aa_complex_o(aa_com5808801942499186519plex_o(R2,X),Y4) ) )
           => ( A4 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1097_finite__transitivity__chain,axiom,
    ! [A4: set_Code_integer,R2: fun_Co2081214437782444048eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(R2,X),X)
       => ( ! [X: code_integer,Y3: code_integer,Z3: code_integer] :
              ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R2,X),Y3)
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(R2,Y3),Z3)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(R2,X),Z3) ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
               => ? [Y4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y4),A4)
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(R2,X),Y4) ) )
           => ( A4 = bot_bo3990330152332043303nteger ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1098_finite__transitivity__chain,axiom,
    ! [A4: set_nat,R2: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R2,X),X)
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R2,X),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R2,Y3),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R2,X),Z3) ) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),A4)
               => ? [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),A4)
                    & aa_nat_o(aa_nat_fun_nat_o(R2,X),Y4) ) )
           => ( A4 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1099_finite__transitivity__chain,axiom,
    ! [A4: set_int,R2: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X: int] : ~ aa_int_o(aa_int_fun_int_o(R2,X),X)
       => ( ! [X: int,Y3: int,Z3: int] :
              ( aa_int_o(aa_int_fun_int_o(R2,X),Y3)
             => ( aa_int_o(aa_int_fun_int_o(R2,Y3),Z3)
               => aa_int_o(aa_int_fun_int_o(R2,X),Z3) ) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),A4)
               => ? [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),A4)
                    & aa_int_o(aa_int_fun_int_o(R2,X),Y4) ) )
           => ( A4 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1100_finite__transitivity__chain,axiom,
    ! [A4: set_o,R2: fun_o_fun_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [X: $o] : ~ aa_o_o(aa_o_fun_o_o(R2,(X)),(X))
       => ( ! [X: $o,Y3: $o,Z3: $o] :
              ( aa_o_o(aa_o_fun_o_o(R2,(X)),(Y3))
             => ( aa_o_o(aa_o_fun_o_o(R2,(Y3)),(Z3))
               => aa_o_o(aa_o_fun_o_o(R2,(X)),(Z3)) ) )
         => ( ! [X: $o] :
                ( aa_set_o_o(member_o((X)),A4)
               => ? [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),A4)
                    & aa_o_o(aa_o_fun_o_o(R2,(X)),(Y4)) ) )
           => ( A4 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_1101_arg__min__least,axiom,
    ! [S2: set_VEBT_VEBT,Y: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( ( S2 != bot_bo8194388402131092736T_VEBT )
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,lattic6139528642216935859BT_rat(F,S2))),aa_VEBT_VEBT_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1102_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( ( S2 != bot_bot_set_real )
       => ( aa_set_real_o(member_real(Y),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S2))),aa_real_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1103_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Y),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2))),aa_complex_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1104_arg__min__least,axiom,
    ! [S2: set_Code_integer,Y: code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(Y),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,lattic5439806495466278992er_rat(F,S2))),aa_Code_integer_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1105_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Y),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2))),aa_nat_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1106_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Y),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2))),aa_int_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1107_arg__min__least,axiom,
    ! [S2: set_o,Y: $o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ( S2 != bot_bot_set_o )
       => ( aa_set_o_o(member_o((Y)),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,aa_set_o_o(lattic2140725968369957399_o_rat(F),S2))),aa_o_rat(F,(Y))) ) ) ) ).

% arg_min_least
tff(fact_1108_arg__min__least,axiom,
    ! [S2: set_VEBT_VEBT,Y: vEBT_VEBT,F: fun_VEBT_VEBT_num] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( ( S2 != bot_bo8194388402131092736T_VEBT )
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y),S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_VEBT_VEBT_num(F,lattic3331990488459210229BT_num(F,S2))),aa_VEBT_VEBT_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1109_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( ( S2 != bot_bot_set_real )
       => ( aa_set_real_o(member_real(Y),S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,lattic1613168225601753569al_num(F,S2))),aa_real_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1110_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Y),S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S2))),aa_complex_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1111_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,T),Xa)),one_one_nat) ) ) ).

% minNull_delete_time_bound'
tff(fact_1112_vebt__member__code_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Leaf((A),(B))),Xa)
    <=> $ite(
          Xa = zero_zero_nat,
          (A),
          $ite(Xa = one_one_nat,(B),$false) ) ) ).

% vebt_member_code(1)
tff(fact_1113_vebt__delete__code_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(vEBT_Leaf((A),(B))),Xa) = $ite(
          Xa = zero_zero_nat,
          vEBT_Leaf($false,(B)),
          $ite(Xa = one_one_nat,vEBT_Leaf((A),$false),vEBT_Leaf((A),(B))) ) ) ).

% vebt_delete_code(1)
tff(fact_1114_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_1115_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_1116_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_1117_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_1118_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_1119_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_1120_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_1121_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_1122_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_1123_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_1124_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_1125_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_1126_zero__less__one__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% zero_less_one_class.zero_le_one
tff(fact_1127_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_1128_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_1129_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_1130_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1131_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1132_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1133_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1134_not__one__le__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).

% not_one_le_zero
tff(fact_1135_not__one__le__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_1136_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_1137_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_1138_bot__empty__eq,axiom,
    ! [X3: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(bot_bo6217218057720408379VEBT_o,X3)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT) ) ).

% bot_empty_eq
tff(fact_1139_bot__empty__eq,axiom,
    ! [X3: real] :
      ( aa_real_o(bot_bot_fun_real_o,X3)
    <=> aa_set_real_o(member_real(X3),bot_bot_set_real) ) ).

% bot_empty_eq
tff(fact_1140_bot__empty__eq,axiom,
    ! [X3: set_nat] :
      ( aa_set_nat_o(bot_bo3967814874310145751_nat_o,X3)
    <=> aa_set_set_nat_o(member_set_nat(X3),bot_bot_set_set_nat) ) ).

% bot_empty_eq
tff(fact_1141_bot__empty__eq,axiom,
    ! [X3: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X3)
    <=> aa_set_nat_o(member_nat(X3),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_1142_bot__empty__eq,axiom,
    ! [X3: int] :
      ( aa_int_o(bot_bot_fun_int_o,X3)
    <=> aa_set_int_o(member_int(X3),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_1143_bot__empty__eq,axiom,
    ! [X3: $o] :
      ( aa_o_o(bot_bot_fun_o_o,(X3))
    <=> aa_set_o_o(member_o((X3)),bot_bot_set_o) ) ).

% bot_empty_eq
tff(fact_1144_Collect__empty__eq__bot,axiom,
    ! [P2: fun_complex_o] :
      ( ( collect_complex(P2) = bot_bot_set_complex )
    <=> ( P2 = bot_bo8750099748576881071plex_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1145_Collect__empty__eq__bot,axiom,
    ! [P2: fun_list_nat_o] :
      ( ( collect_list_nat(P2) = bot_bot_set_list_nat )
    <=> ( P2 = bot_bo8828579575001852285_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1146_Collect__empty__eq__bot,axiom,
    ! [P2: fun_set_nat_o] :
      ( ( collect_set_nat(P2) = bot_bot_set_set_nat )
    <=> ( P2 = bot_bo3967814874310145751_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1147_Collect__empty__eq__bot,axiom,
    ! [P2: fun_nat_o] :
      ( ( collect_nat(P2) = bot_bot_set_nat )
    <=> ( P2 = bot_bot_fun_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1148_Collect__empty__eq__bot,axiom,
    ! [P2: fun_int_o] :
      ( ( collect_int(P2) = bot_bot_set_int )
    <=> ( P2 = bot_bot_fun_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1149_Collect__empty__eq__bot,axiom,
    ! [P2: fun_o_o] :
      ( ( collect_o(P2) = bot_bot_set_o )
    <=> ( P2 = bot_bot_fun_o_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1150_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_1151_linorder__neqE__linordered__idom,axiom,
    ! [Xa: real,Y: real] :
      ( ( Xa != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1152_linorder__neqE__linordered__idom,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( Xa != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1153_linorder__neqE__linordered__idom,axiom,
    ! [Xa: int,Y: int] :
      ( ( Xa != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xa) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1154_subrelI,axiom,
    ! [R: set_Pr1773385645901665561uint32,Sa: set_Pr1773385645901665561uint32] :
      ( ! [X: uint32,Y3: uint32] :
          ( member8027108493173000802uint32(produc1400373151660368625uint32(X,Y3),R)
         => member8027108493173000802uint32(produc1400373151660368625uint32(X,Y3),Sa) )
     => ord_le6429528607962791097uint32(R,Sa) ) ).

% subrelI
tff(fact_1155_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Sa: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3),R)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3),Sa) )
     => ord_le3146513528884898305at_nat(R,Sa) ) ).

% subrelI
tff(fact_1156_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,Sa: set_Pr958786334691620121nt_int] :
      ( ! [X: int,Y3: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(X,Y3),R)
         => member5262025264175285858nt_int(product_Pair_int_int(X,Y3),Sa) )
     => ord_le2843351958646193337nt_int(R,Sa) ) ).

% subrelI
tff(fact_1157_subrelI,axiom,
    ! [R: set_Pr1312406793078187350et_nat,Sa: set_Pr1312406793078187350et_nat] :
      ( ! [X: fun_Pr6460944699146269935_nat_o,Y3: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(produc6017135503889078682et_nat(X,Y3),R)
         => member4513763875858039863et_nat(produc6017135503889078682et_nat(X,Y3),Sa) )
     => ord_le3919382008627836342et_nat(R,Sa) ) ).

% subrelI
tff(fact_1158_subrelI,axiom,
    ! [R: set_Pr4040394263813967220et_nat,Sa: set_Pr4040394263813967220et_nat] :
      ( ! [X: fun_Pr6460944699146269935_nat_o,Y3: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(produc7291855603632349880et_nat(X,Y3),R)
         => member9174161442226468181et_nat(produc7291855603632349880et_nat(X,Y3),Sa) )
     => ord_le5189643068899110356et_nat(R,Sa) ) ).

% subrelI
tff(fact_1159_mod__emp__simp,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).

% mod_emp_simp
tff(fact_1160_le__numeral__extra_I3_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).

% le_numeral_extra(3)
tff(fact_1161_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_1162_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_1163_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_1164_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_1165_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_1166_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_1167_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_1168_zero__neq__one,axiom,
    zero_zero_uint32 != one_one_uint32 ).

% zero_neq_one
tff(fact_1169_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1170_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1171_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1172_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1173_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_1174_le__numeral__extra_I4_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_1175_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_1176_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_1177_less__numeral__extra_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).

% less_numeral_extra(4)
tff(fact_1178_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_1179_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_1180_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_1181_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_1182_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,vEBT_Leaf((A),(B))),zero_zero_nat) = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_1183_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Na: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Na))) = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_1184_nat__descend__induct,axiom,
    ! [Na: nat,P2: fun_nat_o,M: nat] :
      ( ! [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K2)
         => aa_nat_o(P2,K2) )
     => ( ! [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Na)
           => ( ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I4)
                 => aa_nat_o(P2,I4) )
             => aa_nat_o(P2,K2) ) )
       => aa_nat_o(P2,M) ) ) ).

% nat_descend_induct
tff(fact_1185_exists__least__lemma,axiom,
    ! [P2: fun_nat_o] :
      ( ~ aa_nat_o(P2,zero_zero_nat)
     => ( ? [X_13: nat] : aa_nat_o(P2,X_13)
       => ? [N: nat] :
            ( ~ aa_nat_o(P2,N)
            & aa_nat_o(P2,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_1186_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero_nat )
     => ( ( Xa != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : ( Xa != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_1187_list__decode_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero_nat )
     => ~ ! [N: nat] : ( Xa != aa_nat_nat(suc,N) ) ) ).

% list_decode.cases
tff(fact_1188_mod__h__bot__iff_I4_J,axiom,
    ! [Q3: array_VEBT_VEBTi,Y: list_VEBT_VEBTi,H2: heap_e7401611519738050253t_unit] : ~ aa_Pro653082453696054616_nat_o(rep_assn(snga_assn_VEBT_VEBTi(Q3,Y)),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).

% mod_h_bot_iff(4)
tff(fact_1189_complete__interval,axiom,
    ! [A: real,B: real,P2: fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(P2,A)
       => ( ~ aa_real_o(P2,B)
         => ? [C: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
              & ! [X3: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),C) )
                 => aa_real_o(P2,X3) )
              & ! [D3: real] :
                  ( ! [X: real] :
                      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
                        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),D3) )
                     => aa_real_o(P2,X) )
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D3),C) ) ) ) ) ) ).

% complete_interval
tff(fact_1190_complete__interval,axiom,
    ! [A: nat,B: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(P2,A)
       => ( ~ aa_nat_o(P2,B)
         => ? [C: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
              & ! [X3: nat] :
                  ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X3)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),C) )
                 => aa_nat_o(P2,X3) )
              & ! [D3: nat] :
                  ( ! [X: nat] :
                      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),D3) )
                     => aa_nat_o(P2,X) )
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D3),C) ) ) ) ) ) ).

% complete_interval
tff(fact_1191_complete__interval,axiom,
    ! [A: int,B: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(P2,A)
       => ( ~ aa_int_o(P2,B)
         => ? [C: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
              & ! [X3: int] :
                  ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X3)
                    & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),C) )
                 => aa_int_o(P2,X3) )
              & ! [D3: int] :
                  ( ! [X: int] :
                      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
                        & aa_int_o(aa_int_fun_int_o(ord_less_int,X),D3) )
                     => aa_int_o(P2,X) )
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D3),C) ) ) ) ) ) ).

% complete_interval
tff(fact_1192_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1193_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1194_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1195_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1196_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_1197_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(A),A) ).

% verit_comp_simplify1(2)
tff(fact_1198_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1199_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1200_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1201_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_1202_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_1203_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_1204_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_1205_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_1206_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1207_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1208_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1209_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1210_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_1211_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B3)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,B3),A) ) ).

% ex_gt_or_lt
tff(fact_1212_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X3) ) ).

% minf(8)
tff(fact_1213_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X3) ) ).

% minf(8)
tff(fact_1214_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T),X3) ) ).

% minf(8)
tff(fact_1215_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X3) ) ).

% minf(8)
tff(fact_1216_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X3) ) ).

% minf(8)
tff(fact_1217_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),T) ) ).

% minf(6)
tff(fact_1218_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),T) ) ).

% minf(6)
tff(fact_1219_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),T) ) ).

% minf(6)
tff(fact_1220_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),T) ) ).

% minf(6)
tff(fact_1221_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),T) ) ).

% minf(6)
tff(fact_1222_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X3) ) ).

% pinf(8)
tff(fact_1223_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X3) ) ).

% pinf(8)
tff(fact_1224_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T),X3) ) ).

% pinf(8)
tff(fact_1225_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X3) ) ).

% pinf(8)
tff(fact_1226_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X3) ) ).

% pinf(8)
tff(fact_1227_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),T) ) ).

% pinf(6)
tff(fact_1228_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),T) ) ).

% pinf(6)
tff(fact_1229_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),T) ) ).

% pinf(6)
tff(fact_1230_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),T) ) ).

% pinf(6)
tff(fact_1231_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),T) ) ).

% pinf(6)
tff(fact_1232_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D22)
       => ? [E: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1233_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D22)
       => ? [E: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1234_dbl__inc__simps_I2_J,axiom,
    neg_nu4269007558841261821uint32(zero_zero_uint32) = one_one_uint32 ).

% dbl_inc_simps(2)
tff(fact_1235_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_1236_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_1237_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_1238_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : ( aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero_nat ) ).

% VEBT.size_gen(2)
tff(fact_1239_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_1240_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),Xa) = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_1241_frac__eq,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xa) = Xa )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% frac_eq
tff(fact_1242_frac__eq,axiom,
    ! [Xa: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xa) = Xa )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),one_one_rat) ) ) ).

% frac_eq
tff(fact_1243_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Na ) ) ).

% Suc_diff_1
tff(fact_1244_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : ( aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1245_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: uint32] : ( minus_minus_uint32(A,A) = zero_zero_uint32 ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1246_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : ( aa_real_real(minus_minus_real(A),A) = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1247_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1248_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1249_diff__zero,axiom,
    ! [A: rat] : ( aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ) ).

% diff_zero
tff(fact_1250_diff__zero,axiom,
    ! [A: uint32] : ( minus_minus_uint32(A,zero_zero_uint32) = A ) ).

% diff_zero
tff(fact_1251_diff__zero,axiom,
    ! [A: real] : ( aa_real_real(minus_minus_real(A),zero_zero_real) = A ) ).

% diff_zero
tff(fact_1252_diff__zero,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ) ).

% diff_zero
tff(fact_1253_diff__zero,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ) ).

% diff_zero
tff(fact_1254_zero__diff,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ) ).

% zero_diff
tff(fact_1255_diff__0__right,axiom,
    ! [A: rat] : ( aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ) ).

% diff_0_right
tff(fact_1256_diff__0__right,axiom,
    ! [A: uint32] : ( minus_minus_uint32(A,zero_zero_uint32) = A ) ).

% diff_0_right
tff(fact_1257_diff__0__right,axiom,
    ! [A: real] : ( aa_real_real(minus_minus_real(A),zero_zero_real) = A ) ).

% diff_0_right
tff(fact_1258_diff__0__right,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ) ).

% diff_0_right
tff(fact_1259_diff__self,axiom,
    ! [A: rat] : ( aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ) ).

% diff_self
tff(fact_1260_diff__self,axiom,
    ! [A: uint32] : ( minus_minus_uint32(A,A) = zero_zero_uint32 ) ).

% diff_self
tff(fact_1261_diff__self,axiom,
    ! [A: real] : ( aa_real_real(minus_minus_real(A),A) = zero_zero_real ) ).

% diff_self
tff(fact_1262_diff__self,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ) ).

% diff_self
tff(fact_1263_Suc__diff__diff,axiom,
    ! [M: nat,Na: 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)),Na)),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),Na)),K) ) ).

% Suc_diff_diff
tff(fact_1264_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ).

% diff_Suc_Suc
tff(fact_1265_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_1266_diff__0__eq__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Na) = zero_zero_nat ) ).

% diff_0_eq_0
tff(fact_1267_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1268_zero__comp__diff__simps_I1_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_1269_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_1270_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% zero_comp_diff_simps(1)
tff(fact_1271_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1272_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1273_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1274_zero__comp__diff__simps_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_1275_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_1276_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% zero_comp_diff_simps(2)
tff(fact_1277_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1278_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1279_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1280_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1281_diff__numeral__special_I9_J,axiom,
    minus_minus_uint32(one_one_uint32,one_one_uint32) = zero_zero_uint32 ).

% diff_numeral_special(9)
tff(fact_1282_diff__numeral__special_I9_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_1283_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_1284_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% zero_less_diff
tff(fact_1285_diff__Suc__1,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),one_one_nat) = Na ) ).

% diff_Suc_1
tff(fact_1286_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1287_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% diff_is_0_eq
tff(fact_1288_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% Suc_pred
tff(fact_1289_Suc__diff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_1290_diff__eq__diff__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32,D: uint32] :
      ( ( minus_minus_uint32(A,B) = minus_minus_uint32(C2,D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1291_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1292_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D) )
     => ( ( A = B )
      <=> ( C2 = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1293_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: uint32,C2: uint32,B: uint32] : ( minus_minus_uint32(minus_minus_uint32(A,C2),B) = minus_minus_uint32(minus_minus_uint32(A,B),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1294_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C2: real,B: real] : ( aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C2)),B) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1295_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C2: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C2)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1296_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C2: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1297_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_1298_le__some__optE,axiom,
    ! [M: set_int,Xa: option_set_int] :
      ( ord_le353528952715127954et_int(some_set_int(M),Xa)
     => ~ ! [M8: set_int] :
            ( ( Xa = some_set_int(M8) )
           => ~ aa_set_int_o(ord_less_eq_set_int(M),M8) ) ) ).

% le_some_optE
tff(fact_1299_le__some__optE,axiom,
    ! [M: rat,Xa: option_rat] :
      ( ord_le2406147912482264968on_rat(some_rat(M),Xa)
     => ~ ! [M8: rat] :
            ( ( Xa = some_rat(M8) )
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),M8) ) ) ).

% le_some_optE
tff(fact_1300_le__some__optE,axiom,
    ! [M: num,Xa: option_num] :
      ( ord_le6622620407824499402on_num(some_num(M),Xa)
     => ~ ! [M8: num] :
            ( ( Xa = some_num(M8) )
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),M8) ) ) ).

% le_some_optE
tff(fact_1301_le__some__optE,axiom,
    ! [M: nat,Xa: option_nat] :
      ( ord_le5914376470875661696on_nat(some_nat(M),Xa)
     => ~ ! [M8: nat] :
            ( ( Xa = some_nat(M8) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),M8) ) ) ).

% le_some_optE
tff(fact_1302_le__some__optE,axiom,
    ! [M: int,Xa: option_int] :
      ( ord_le1736525451366464988on_int(some_int(M),Xa)
     => ~ ! [M8: int] :
            ( ( Xa = some_int(M8) )
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),M8) ) ) ).

% le_some_optE
tff(fact_1303_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_1304_eq__iff__diff__eq__0,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A = B )
    <=> ( minus_minus_uint32(A,B) = zero_zero_uint32 ) ) ).

% eq_iff_diff_eq_0
tff(fact_1305_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(minus_minus_real(A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_1306_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_1307_diff__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C2)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_mono
tff(fact_1308_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_mono
tff(fact_1309_diff__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_1310_diff__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(C2),A)),aa_real_real(minus_minus_real(C2),B)) ) ).

% diff_left_mono
tff(fact_1311_diff__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C2),A)),aa_rat_rat(minus_minus_rat(C2),B)) ) ).

% diff_left_mono
tff(fact_1312_diff__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)) ) ).

% diff_left_mono
tff(fact_1313_diff__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C2)),aa_real_real(minus_minus_real(B),C2)) ) ).

% diff_right_mono
tff(fact_1314_diff__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),C2)) ) ).

% diff_right_mono
tff(fact_1315_diff__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) ) ).

% diff_right_mono
tff(fact_1316_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C2),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1317_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C2),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1318_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1319_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C2)),aa_real_real(minus_minus_real(B),C2)) ) ).

% diff_strict_right_mono
tff(fact_1320_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),C2)) ) ).

% diff_strict_right_mono
tff(fact_1321_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) ) ).

% diff_strict_right_mono
tff(fact_1322_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(C2),A)),aa_real_real(minus_minus_real(C2),B)) ) ).

% diff_strict_left_mono
tff(fact_1323_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C2),A)),aa_rat_rat(minus_minus_rat(C2),B)) ) ).

% diff_strict_left_mono
tff(fact_1324_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B)) ) ).

% diff_strict_left_mono
tff(fact_1325_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C2),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D) ) ) ).

% diff_eq_diff_less
tff(fact_1326_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C2),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D) ) ) ).

% diff_eq_diff_less
tff(fact_1327_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D) ) ) ).

% diff_eq_diff_less
tff(fact_1328_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C2)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_strict_mono
tff(fact_1329_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C2)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_strict_mono
tff(fact_1330_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_1331_zero__induct__lemma,axiom,
    ! [P2: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P2,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P2,aa_nat_nat(suc,N))
           => aa_nat_o(P2,N) )
       => aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_1332_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M) = zero_zero_nat )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_1333_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_1334_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_1335_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_1336_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_1337_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% le_diff_iff
tff(fact_1338_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( 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,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1339_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),L)) ) ).

% diff_le_mono
tff(fact_1340_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),M) ).

% diff_le_self
tff(fact_1341_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_1342_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ).

% diff_le_mono2
tff(fact_1343_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList),Summary)),Xa) = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_1344_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_1345_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_1346_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_1347_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_1348_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_1349_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_1350_Suc__to__right,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(suc,Na) = M )
     => ( Na = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_1351_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_1352_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_1353_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),M) ) ) ).

% diff_less
tff(fact_1354_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% Suc_diff_le
tff(fact_1355_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na)) = 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)),Na) ) ).

% diff_Suc_eq_diff_pred
tff(fact_1356_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ) ).

% less_diff_iff
tff(fact_1357_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C2)) ) ) ).

% diff_less_mono
tff(fact_1358_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary)),Xa) = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_1359_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb),Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_1360_frac__ge__0,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xa)) ).

% frac_ge_0
tff(fact_1361_frac__ge__0,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,Xa)) ).

% frac_ge_0
tff(fact_1362_frac__lt__1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,Xa)),one_one_real) ).

% frac_lt_1
tff(fact_1363_frac__lt__1,axiom,
    ! [Xa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(archimedean_frac_rat,Xa)),one_one_rat) ).

% frac_lt_1
tff(fact_1364_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_1365_pinf_I1_J,axiom,
    ! [P2: fun_real_o,P4: fun_real_o,Q2: fun_real_o,Q4: fun_real_o] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
         => ( aa_real_o(P2,X)
          <=> aa_real_o(P4,X) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
           => ( aa_real_o(Q2,X)
            <=> aa_real_o(Q4,X) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( ( aa_real_o(P2,X3)
                & aa_real_o(Q2,X3) )
            <=> ( aa_real_o(P4,X3)
                & aa_real_o(Q4,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1366_pinf_I1_J,axiom,
    ! [P2: fun_rat_o,P4: fun_rat_o,Q2: fun_rat_o,Q4: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
         => ( aa_rat_o(P2,X)
          <=> aa_rat_o(P4,X) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
           => ( aa_rat_o(Q2,X)
            <=> aa_rat_o(Q4,X) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( ( aa_rat_o(P2,X3)
                & aa_rat_o(Q2,X3) )
            <=> ( aa_rat_o(P4,X3)
                & aa_rat_o(Q4,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1367_pinf_I1_J,axiom,
    ! [P2: fun_num_o,P4: fun_num_o,Q2: fun_num_o,Q4: fun_num_o] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
         => ( aa_num_o(P2,X)
          <=> aa_num_o(P4,X) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
           => ( aa_num_o(Q2,X)
            <=> aa_num_o(Q4,X) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
           => ( ( aa_num_o(P2,X3)
                & aa_num_o(Q2,X3) )
            <=> ( aa_num_o(P4,X3)
                & aa_num_o(Q4,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1368_pinf_I1_J,axiom,
    ! [P2: fun_nat_o,P4: fun_nat_o,Q2: fun_nat_o,Q4: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
         => ( aa_nat_o(P2,X)
          <=> aa_nat_o(P4,X) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
           => ( aa_nat_o(Q2,X)
            <=> aa_nat_o(Q4,X) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( ( aa_nat_o(P2,X3)
                & aa_nat_o(Q2,X3) )
            <=> ( aa_nat_o(P4,X3)
                & aa_nat_o(Q4,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1369_pinf_I1_J,axiom,
    ! [P2: fun_int_o,P4: fun_int_o,Q2: fun_int_o,Q4: fun_int_o] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
         => ( aa_int_o(P2,X)
          <=> aa_int_o(P4,X) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
           => ( aa_int_o(Q2,X)
            <=> aa_int_o(Q4,X) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( ( aa_int_o(P2,X3)
                & aa_int_o(Q2,X3) )
            <=> ( aa_int_o(P4,X3)
                & aa_int_o(Q4,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1370_pinf_I2_J,axiom,
    ! [P2: fun_real_o,P4: fun_real_o,Q2: fun_real_o,Q4: fun_real_o] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
         => ( aa_real_o(P2,X)
          <=> aa_real_o(P4,X) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
           => ( aa_real_o(Q2,X)
            <=> aa_real_o(Q4,X) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( ( aa_real_o(P2,X3)
                | aa_real_o(Q2,X3) )
            <=> ( aa_real_o(P4,X3)
                | aa_real_o(Q4,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1371_pinf_I2_J,axiom,
    ! [P2: fun_rat_o,P4: fun_rat_o,Q2: fun_rat_o,Q4: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
         => ( aa_rat_o(P2,X)
          <=> aa_rat_o(P4,X) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
           => ( aa_rat_o(Q2,X)
            <=> aa_rat_o(Q4,X) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( ( aa_rat_o(P2,X3)
                | aa_rat_o(Q2,X3) )
            <=> ( aa_rat_o(P4,X3)
                | aa_rat_o(Q4,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1372_pinf_I2_J,axiom,
    ! [P2: fun_num_o,P4: fun_num_o,Q2: fun_num_o,Q4: fun_num_o] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
         => ( aa_num_o(P2,X)
          <=> aa_num_o(P4,X) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
           => ( aa_num_o(Q2,X)
            <=> aa_num_o(Q4,X) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
           => ( ( aa_num_o(P2,X3)
                | aa_num_o(Q2,X3) )
            <=> ( aa_num_o(P4,X3)
                | aa_num_o(Q4,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1373_pinf_I2_J,axiom,
    ! [P2: fun_nat_o,P4: fun_nat_o,Q2: fun_nat_o,Q4: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
         => ( aa_nat_o(P2,X)
          <=> aa_nat_o(P4,X) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
           => ( aa_nat_o(Q2,X)
            <=> aa_nat_o(Q4,X) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( ( aa_nat_o(P2,X3)
                | aa_nat_o(Q2,X3) )
            <=> ( aa_nat_o(P4,X3)
                | aa_nat_o(Q4,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1374_pinf_I2_J,axiom,
    ! [P2: fun_int_o,P4: fun_int_o,Q2: fun_int_o,Q4: fun_int_o] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
         => ( aa_int_o(P2,X)
          <=> aa_int_o(P4,X) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
           => ( aa_int_o(Q2,X)
            <=> aa_int_o(Q4,X) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( ( aa_int_o(P2,X3)
                | aa_int_o(Q2,X3) )
            <=> ( aa_int_o(P4,X3)
                | aa_int_o(Q4,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1375_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1376_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1377_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1378_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1379_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(3)
tff(fact_1380_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1381_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1382_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1383_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1384_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( X3 != T ) ) ).

% pinf(4)
tff(fact_1385_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),T) ) ).

% pinf(5)
tff(fact_1386_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),T) ) ).

% pinf(5)
tff(fact_1387_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),T) ) ).

% pinf(5)
tff(fact_1388_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),T) ) ).

% pinf(5)
tff(fact_1389_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),T) ) ).

% pinf(5)
tff(fact_1390_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,T),X3) ) ).

% pinf(7)
tff(fact_1391_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X3) ) ).

% pinf(7)
tff(fact_1392_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,T),X3) ) ).

% pinf(7)
tff(fact_1393_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X3) ) ).

% pinf(7)
tff(fact_1394_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,T),X3) ) ).

% pinf(7)
tff(fact_1395_minf_I1_J,axiom,
    ! [P2: fun_real_o,P4: fun_real_o,Q2: fun_real_o,Q4: fun_real_o] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
         => ( aa_real_o(P2,X)
          <=> aa_real_o(P4,X) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
           => ( aa_real_o(Q2,X)
            <=> aa_real_o(Q4,X) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( ( aa_real_o(P2,X3)
                & aa_real_o(Q2,X3) )
            <=> ( aa_real_o(P4,X3)
                & aa_real_o(Q4,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1396_minf_I1_J,axiom,
    ! [P2: fun_rat_o,P4: fun_rat_o,Q2: fun_rat_o,Q4: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
         => ( aa_rat_o(P2,X)
          <=> aa_rat_o(P4,X) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
           => ( aa_rat_o(Q2,X)
            <=> aa_rat_o(Q4,X) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( ( aa_rat_o(P2,X3)
                & aa_rat_o(Q2,X3) )
            <=> ( aa_rat_o(P4,X3)
                & aa_rat_o(Q4,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1397_minf_I1_J,axiom,
    ! [P2: fun_num_o,P4: fun_num_o,Q2: fun_num_o,Q4: fun_num_o] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
         => ( aa_num_o(P2,X)
          <=> aa_num_o(P4,X) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
           => ( aa_num_o(Q2,X)
            <=> aa_num_o(Q4,X) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
           => ( ( aa_num_o(P2,X3)
                & aa_num_o(Q2,X3) )
            <=> ( aa_num_o(P4,X3)
                & aa_num_o(Q4,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1398_minf_I1_J,axiom,
    ! [P2: fun_nat_o,P4: fun_nat_o,Q2: fun_nat_o,Q4: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
         => ( aa_nat_o(P2,X)
          <=> aa_nat_o(P4,X) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
           => ( aa_nat_o(Q2,X)
            <=> aa_nat_o(Q4,X) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( ( aa_nat_o(P2,X3)
                & aa_nat_o(Q2,X3) )
            <=> ( aa_nat_o(P4,X3)
                & aa_nat_o(Q4,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1399_minf_I1_J,axiom,
    ! [P2: fun_int_o,P4: fun_int_o,Q2: fun_int_o,Q4: fun_int_o] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
         => ( aa_int_o(P2,X)
          <=> aa_int_o(P4,X) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
           => ( aa_int_o(Q2,X)
            <=> aa_int_o(Q4,X) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( ( aa_int_o(P2,X3)
                & aa_int_o(Q2,X3) )
            <=> ( aa_int_o(P4,X3)
                & aa_int_o(Q4,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1400_minf_I2_J,axiom,
    ! [P2: fun_real_o,P4: fun_real_o,Q2: fun_real_o,Q4: fun_real_o] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
         => ( aa_real_o(P2,X)
          <=> aa_real_o(P4,X) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
           => ( aa_real_o(Q2,X)
            <=> aa_real_o(Q4,X) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( ( aa_real_o(P2,X3)
                | aa_real_o(Q2,X3) )
            <=> ( aa_real_o(P4,X3)
                | aa_real_o(Q4,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1401_minf_I2_J,axiom,
    ! [P2: fun_rat_o,P4: fun_rat_o,Q2: fun_rat_o,Q4: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
         => ( aa_rat_o(P2,X)
          <=> aa_rat_o(P4,X) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
           => ( aa_rat_o(Q2,X)
            <=> aa_rat_o(Q4,X) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( ( aa_rat_o(P2,X3)
                | aa_rat_o(Q2,X3) )
            <=> ( aa_rat_o(P4,X3)
                | aa_rat_o(Q4,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1402_minf_I2_J,axiom,
    ! [P2: fun_num_o,P4: fun_num_o,Q2: fun_num_o,Q4: fun_num_o] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
         => ( aa_num_o(P2,X)
          <=> aa_num_o(P4,X) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
           => ( aa_num_o(Q2,X)
            <=> aa_num_o(Q4,X) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
           => ( ( aa_num_o(P2,X3)
                | aa_num_o(Q2,X3) )
            <=> ( aa_num_o(P4,X3)
                | aa_num_o(Q4,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1403_minf_I2_J,axiom,
    ! [P2: fun_nat_o,P4: fun_nat_o,Q2: fun_nat_o,Q4: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
         => ( aa_nat_o(P2,X)
          <=> aa_nat_o(P4,X) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
           => ( aa_nat_o(Q2,X)
            <=> aa_nat_o(Q4,X) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( ( aa_nat_o(P2,X3)
                | aa_nat_o(Q2,X3) )
            <=> ( aa_nat_o(P4,X3)
                | aa_nat_o(Q4,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1404_minf_I2_J,axiom,
    ! [P2: fun_int_o,P4: fun_int_o,Q2: fun_int_o,Q4: fun_int_o] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
         => ( aa_int_o(P2,X)
          <=> aa_int_o(P4,X) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
           => ( aa_int_o(Q2,X)
            <=> aa_int_o(Q4,X) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( ( aa_int_o(P2,X3)
                | aa_int_o(Q2,X3) )
            <=> ( aa_int_o(P4,X3)
                | aa_int_o(Q4,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1405_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1406_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1407_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1408_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1409_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( X3 != T ) ) ).

% minf(3)
tff(fact_1410_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1411_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1412_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1413_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1414_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( X3 != T ) ) ).

% minf(4)
tff(fact_1415_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),T) ) ).

% minf(5)
tff(fact_1416_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),T) ) ).

% minf(5)
tff(fact_1417_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),T) ) ).

% minf(5)
tff(fact_1418_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),T) ) ).

% minf(5)
tff(fact_1419_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),T) ) ).

% minf(5)
tff(fact_1420_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,T),X3) ) ).

% minf(7)
tff(fact_1421_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X3) ) ).

% minf(7)
tff(fact_1422_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,T),X3) ) ).

% minf(7)
tff(fact_1423_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X3) ) ).

% minf(7)
tff(fact_1424_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,T),X3) ) ).

% minf(7)
tff(fact_1425_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,Xa: nat] : ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst),Smry)),Xa) = aa_VEB5832953974749077948T_VEBT(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_1426_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz)),Xa) ).

% vebt_member.simps(3)
tff(fact_1427_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( ! [Uv: $o] : ( Xa != vEBT_Leaf($true,(Uv)) )
       => ( ! [Uu: $o] : ( Xa != vEBT_Leaf((Uu),$true) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1428_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va2),Vb)),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1429_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M) ) ) ).

% nz_le_conv_less
tff(fact_1430_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_1431_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1432_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_1433_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,Xa: nat] : ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm)),Xa) = aa_VEB5832953974749077948T_VEBT(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_1434_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),Xa) ).

% vebt_member.simps(4)
tff(fact_1435_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList),Summary)),Xa) = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_1436_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Na: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Na))) = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_1437_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,vEBT_Leaf((A),(B))),zero_zero_nat) = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_1438_less__option__Some,axiom,
    ! [Xa: real,Y: real] :
      ( ord_less_option_real(some_real(Xa),some_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% less_option_Some
tff(fact_1439_less__option__Some,axiom,
    ! [Xa: rat,Y: rat] :
      ( ord_less_option_rat(some_rat(Xa),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ).

% less_option_Some
tff(fact_1440_less__option__Some,axiom,
    ! [Xa: num,Y: num] :
      ( ord_less_option_num(some_num(Xa),some_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Y) ) ).

% less_option_Some
tff(fact_1441_less__option__Some,axiom,
    ! [Xa: nat,Y: nat] :
      ( ord_less_option_nat(some_nat(Xa),some_nat(Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ).

% less_option_Some
tff(fact_1442_less__option__Some,axiom,
    ! [Xa: int,Y: int] :
      ( ord_less_option_int(some_int(Xa),some_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ).

% less_option_Some
tff(fact_1443_less__eq__option__Some,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ord_le353528952715127954et_int(some_set_int(Xa),some_set_int(Y))
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),Y) ) ).

% less_eq_option_Some
tff(fact_1444_less__eq__option__Some,axiom,
    ! [Xa: rat,Y: rat] :
      ( ord_le2406147912482264968on_rat(some_rat(Xa),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% less_eq_option_Some
tff(fact_1445_less__eq__option__Some,axiom,
    ! [Xa: num,Y: num] :
      ( ord_le6622620407824499402on_num(some_num(Xa),some_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y) ) ).

% less_eq_option_Some
tff(fact_1446_less__eq__option__Some,axiom,
    ! [Xa: nat,Y: nat] :
      ( ord_le5914376470875661696on_nat(some_nat(Xa),some_nat(Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ).

% less_eq_option_Some
tff(fact_1447_less__eq__option__Some,axiom,
    ! [Xa: int,Y: int] :
      ( ord_le1736525451366464988on_int(some_int(Xa),some_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ).

% less_eq_option_Some
tff(fact_1448_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)
     => ( ( Mi = Ma )
       => ( ! [X3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
             => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) )
          & ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1449_diff__shunt__var,axiom,
    ! [Xa: assn,Y: assn] :
      ( ( minus_minus_assn(Xa,Y) = bot_bot_assn )
    <=> ord_less_eq_assn(Xa,Y) ) ).

% diff_shunt_var
tff(fact_1450_diff__shunt__var,axiom,
    ! [Xa: set_o,Y: set_o] :
      ( ( minus_minus_set_o(Xa,Y) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(Xa),Y) ) ).

% diff_shunt_var
tff(fact_1451_diff__shunt__var,axiom,
    ! [Xa: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(Xa,Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),Y) ) ).

% diff_shunt_var
tff(fact_1452_diff__shunt__var,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( ( minus_minus_set_int(Xa,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xa),Y) ) ).

% diff_shunt_var
tff(fact_1453_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      ( nat_prod_decode_aux(K,M) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)),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_1454_prod__decode__aux_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xa)))) ) ) ).

% prod_decode_aux.elims
tff(fact_1455_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Uw) )
     => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Uz2) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2),X) )
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2),X) )
           => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd),X) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1456_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),zero_zero_nat) )
     => ( ! [A3: $o,B3: $o] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,zero_zero_nat)) )
       => ( ! [A3: $o,B3: $o,N: nat] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N))) )
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Uu) )
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2),X) )
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),X) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_1457_diff__diff__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na) ) ) ).

% diff_diff_less
tff(fact_1458_Diff__empty,axiom,
    ! [A4: set_int] : ( minus_minus_set_int(A4,bot_bot_set_int) = A4 ) ).

% Diff_empty
tff(fact_1459_Diff__empty,axiom,
    ! [A4: set_o] : ( minus_minus_set_o(A4,bot_bot_set_o) = A4 ) ).

% Diff_empty
tff(fact_1460_Diff__empty,axiom,
    ! [A4: set_nat] : ( minus_minus_set_nat(A4,bot_bot_set_nat) = A4 ) ).

% Diff_empty
tff(fact_1461_empty__Diff,axiom,
    ! [A4: set_int] : ( minus_minus_set_int(bot_bot_set_int,A4) = bot_bot_set_int ) ).

% empty_Diff
tff(fact_1462_empty__Diff,axiom,
    ! [A4: set_o] : ( minus_minus_set_o(bot_bot_set_o,A4) = bot_bot_set_o ) ).

% empty_Diff
tff(fact_1463_empty__Diff,axiom,
    ! [A4: set_nat] : ( minus_minus_set_nat(bot_bot_set_nat,A4) = bot_bot_set_nat ) ).

% empty_Diff
tff(fact_1464_Diff__cancel,axiom,
    ! [A4: set_int] : ( minus_minus_set_int(A4,A4) = bot_bot_set_int ) ).

% Diff_cancel
tff(fact_1465_Diff__cancel,axiom,
    ! [A4: set_o] : ( minus_minus_set_o(A4,A4) = bot_bot_set_o ) ).

% Diff_cancel
tff(fact_1466_Diff__cancel,axiom,
    ! [A4: set_nat] : ( minus_minus_set_nat(A4,A4) = bot_bot_set_nat ) ).

% Diff_cancel
tff(fact_1467_finite__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_set_int_o(finite_finite_int,minus_minus_set_int(A4,B4)) ) ).

% finite_Diff
tff(fact_1468_finite__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,B4)) ) ).

% finite_Diff
tff(fact_1469_finite__Diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A4,B4)) ) ).

% finite_Diff
tff(fact_1470_finite__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A4,B4)) ) ).

% finite_Diff
tff(fact_1471_finite__Diff2,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A4,B4))
      <=> aa_set_int_o(finite_finite_int,A4) ) ) ).

% finite_Diff2
tff(fact_1472_finite__Diff2,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,B4))
      <=> aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% finite_Diff2
tff(fact_1473_finite__Diff2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A4,B4))
      <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ) ).

% finite_Diff2
tff(fact_1474_finite__Diff2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A4,B4))
      <=> aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_Diff2
tff(fact_1475_Diff__eq__empty__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( minus_minus_set_o(A4,B4) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1476_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat(A4,B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1477_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A4,B4) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1478_lesseq__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
    <=> vEBT_VEBT_lesseq(some_nat(Xa),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_1479_less__eq__option__None__code,axiom,
    ! [Xa: option_nat] : ord_le5914376470875661696on_nat(none_nat,Xa) ).

% less_eq_option_None_code
tff(fact_1480_less__option__None,axiom,
    ! [Xa: option_nat] : ~ ord_less_option_nat(Xa,none_nat) ).

% less_option_None
tff(fact_1481_not__Some__eq2,axiom,
    ! [V2: option3972171592325465343uint32] :
      ( ! [X5: uint32,Y5: uint32] : ( V2 != some_P6695178580830101022uint32(produc1400373151660368625uint32(X5,Y5)) )
    <=> ( V2 = none_P7441528354948028570uint32 ) ) ).

% not_Some_eq2
tff(fact_1482_not__Some__eq2,axiom,
    ! [V2: option4624381673175914239nt_int] :
      ( ! [X5: int,Y5: int] : ( V2 != some_P4184893108420464158nt_int(product_Pair_int_int(X5,Y5)) )
    <=> ( V2 = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
tff(fact_1483_not__Some__eq2,axiom,
    ! [V2: option4955102875562834416et_nat] :
      ( ! [X5: fun_Pr6460944699146269935_nat_o,Y5: produc3658429121746597890et_nat] : ( V2 != some_P994283844982041979et_nat(produc6017135503889078682et_nat(X5,Y5)) )
    <=> ( V2 = none_P7891021755743879167et_nat ) ) ).

% not_Some_eq2
tff(fact_1484_not__Some__eq2,axiom,
    ! [V2: option5899113203645146382et_nat] :
      ( ! [X5: fun_Pr6460944699146269935_nat_o,Y5: produc8799533983703829792et_nat] : ( V2 != some_P3006888012174695577et_nat(produc7291855603632349880et_nat(X5,Y5)) )
    <=> ( V2 = none_P2758821087663475997et_nat ) ) ).

% not_Some_eq2
tff(fact_1485_not__Some__eq2,axiom,
    ! [V2: option4927543243414619207at_nat] :
      ( ! [X5: nat,Y5: nat] : ( V2 != some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y5)) )
    <=> ( V2 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
tff(fact_1486_less__eq__option__Some__None,axiom,
    ! [Xa: nat] : ~ ord_le5914376470875661696on_nat(some_nat(Xa),none_nat) ).

% less_eq_option_Some_None
tff(fact_1487_less__option__None__Some__code,axiom,
    ! [Xa: nat] : ord_less_option_nat(none_nat,some_nat(Xa)) ).

% less_option_None_Some_code
tff(fact_1488_less__eq__option__None,axiom,
    ! [Xa: option_nat] : ord_le5914376470875661696on_nat(none_nat,Xa) ).

% less_eq_option_None
tff(fact_1489_less__eq__option__None__is__None,axiom,
    ! [Xa: option_nat] :
      ( ord_le5914376470875661696on_nat(Xa,none_nat)
     => ( Xa = none_nat ) ) ).

% less_eq_option_None_is_None
tff(fact_1490_less__option__None__Some,axiom,
    ! [Xa: nat] : ord_less_option_nat(none_nat,some_nat(Xa)) ).

% less_option_None_Some
tff(fact_1491_less__option__None__is__Some,axiom,
    ! [Xa: option_nat] :
      ( ord_less_option_nat(none_nat,Xa)
     => ? [Z3: nat] : ( Xa = some_nat(Z3) ) ) ).

% less_option_None_is_Some
tff(fact_1492_Diff__infinite__finite,axiom,
    ! [T2: set_int,S2: set_int] :
      ( aa_set_int_o(finite_finite_int,T2)
     => ( ~ aa_set_int_o(finite_finite_int,S2)
       => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S2,T2)) ) ) ).

% Diff_infinite_finite
tff(fact_1493_Diff__infinite__finite,axiom,
    ! [T2: set_complex,S2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,T2)) ) ) ).

% Diff_infinite_finite
tff(fact_1494_Diff__infinite__finite,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(S2,T2)) ) ) ).

% Diff_infinite_finite
tff(fact_1495_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S2,T2)) ) ) ).

% Diff_infinite_finite
tff(fact_1496_Diff__mono,axiom,
    ! [A4: set_nat,C3: set_nat,D4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),C3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D4),B4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A4,B4)),minus_minus_set_nat(C3,D4)) ) ) ).

% Diff_mono
tff(fact_1497_Diff__mono,axiom,
    ! [A4: set_int,C3: set_int,D4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),C3)
     => ( aa_set_int_o(ord_less_eq_set_int(D4),B4)
       => aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,B4)),minus_minus_set_int(C3,D4)) ) ) ).

% Diff_mono
tff(fact_1498_Diff__subset,axiom,
    ! [A4: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A4,B4)),A4) ).

% Diff_subset
tff(fact_1499_Diff__subset,axiom,
    ! [A4: set_int,B4: set_int] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,B4)),A4) ).

% Diff_subset
tff(fact_1500_double__diff,axiom,
    ! [A4: set_nat,B4: set_nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C3)
       => ( minus_minus_set_nat(B4,minus_minus_set_nat(C3,A4)) = A4 ) ) ) ).

% double_diff
tff(fact_1501_double__diff,axiom,
    ! [A4: set_int,B4: set_int,C3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C3)
       => ( minus_minus_set_int(B4,minus_minus_set_int(C3,A4)) = A4 ) ) ) ).

% double_diff
tff(fact_1502_psubset__imp__ex__mem,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A4),B4)
     => ? [B3: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1503_psubset__imp__ex__mem,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_set_real(A4),B4)
     => ? [B3: real] : aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1504_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
     => ? [B3: int] : aa_set_int_o(member_int(B3),minus_minus_set_int(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1505_psubset__imp__ex__mem,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A4),B4)
     => ? [B3: set_nat] : aa_set_set_nat_o(member_set_nat(B3),minus_2163939370556025621et_nat(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1506_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
     => ? [B3: nat] : aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B4,A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1507_subset__minus__empty,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( minus_minus_set_o(A4,B4) = bot_bot_set_o ) ) ).

% subset_minus_empty
tff(fact_1508_subset__minus__empty,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( minus_minus_set_nat(A4,B4) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_1509_subset__minus__empty,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( minus_minus_set_int(A4,B4) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_1510_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] : ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu2) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) ) ).

% vebt_delete.simps(4)
tff(fact_1511_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1512_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xa) ).

% vebt_member.simps(2)
tff(fact_1513_vebt__member__code_I2_J,axiom,
    ! [T: nat,R: list_VEBT_VEBT,E2: vEBT_VEBT,Xa: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,T,R),E2)),Xa) ).

% vebt_member_code(2)
tff(fact_1514_bijective__def,axiom,
    ! [R2: set_Pr1773385645901665561uint32] :
      ( biject2923219584331114343uint32(R2)
    <=> ( ! [X5: uint32,Y5: uint32,Z2: uint32] :
            ( ( member8027108493173000802uint32(produc1400373151660368625uint32(X5,Y5),R2)
              & member8027108493173000802uint32(produc1400373151660368625uint32(X5,Z2),R2) )
           => ( Y5 = Z2 ) )
        & ! [X5: uint32,Y5: uint32,Z2: uint32] :
            ( ( member8027108493173000802uint32(produc1400373151660368625uint32(X5,Z2),R2)
              & member8027108493173000802uint32(produc1400373151660368625uint32(Y5,Z2),R2) )
           => ( X5 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_1515_bijective__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( bijective_nat_nat(R2)
    <=> ( ! [X5: nat,Y5: nat,Z2: nat] :
            ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y5),R2)
              & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Z2),R2) )
           => ( Y5 = Z2 ) )
        & ! [X5: nat,Y5: nat,Z2: nat] :
            ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Z2),R2)
              & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y5),Z2),R2) )
           => ( X5 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_1516_bijective__def,axiom,
    ! [R2: set_Pr958786334691620121nt_int] :
      ( bijective_int_int(R2)
    <=> ( ! [X5: int,Y5: int,Z2: int] :
            ( ( member5262025264175285858nt_int(product_Pair_int_int(X5,Y5),R2)
              & member5262025264175285858nt_int(product_Pair_int_int(X5,Z2),R2) )
           => ( Y5 = Z2 ) )
        & ! [X5: int,Y5: int,Z2: int] :
            ( ( member5262025264175285858nt_int(product_Pair_int_int(X5,Z2),R2)
              & member5262025264175285858nt_int(product_Pair_int_int(Y5,Z2),R2) )
           => ( X5 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_1517_bijective__def,axiom,
    ! [R2: set_Pr1312406793078187350et_nat] :
      ( biject6574446902887564708et_nat(R2)
    <=> ( ! [X5: fun_Pr6460944699146269935_nat_o,Y5: produc3658429121746597890et_nat,Z2: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(produc6017135503889078682et_nat(X5,Y5),R2)
              & member4513763875858039863et_nat(produc6017135503889078682et_nat(X5,Z2),R2) )
           => ( Y5 = Z2 ) )
        & ! [X5: fun_Pr6460944699146269935_nat_o,Y5: fun_Pr6460944699146269935_nat_o,Z2: produc3658429121746597890et_nat] :
            ( ( member4513763875858039863et_nat(produc6017135503889078682et_nat(X5,Z2),R2)
              & member4513763875858039863et_nat(produc6017135503889078682et_nat(Y5,Z2),R2) )
           => ( X5 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_1518_bijective__def,axiom,
    ! [R2: set_Pr4040394263813967220et_nat] :
      ( biject4066639966817638850et_nat(R2)
    <=> ( ! [X5: fun_Pr6460944699146269935_nat_o,Y5: produc8799533983703829792et_nat,Z2: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(produc7291855603632349880et_nat(X5,Y5),R2)
              & member9174161442226468181et_nat(produc7291855603632349880et_nat(X5,Z2),R2) )
           => ( Y5 = Z2 ) )
        & ! [X5: fun_Pr6460944699146269935_nat_o,Y5: fun_Pr6460944699146269935_nat_o,Z2: produc8799533983703829792et_nat] :
            ( ( member9174161442226468181et_nat(produc7291855603632349880et_nat(X5,Z2),R2)
              & member9174161442226468181et_nat(produc7291855603632349880et_nat(Y5,Z2),R2) )
           => ( X5 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_1519_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( ( Xa != vEBT_Leaf($false,$false) )
       => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy) ) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1520_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_1521_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu2) = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_1522_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Uu2) = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_1523_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
      <=> (Y) )
     => ( ( ( Xa = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv: $o] : ( Xa = vEBT_Leaf($true,(Uv)) )
           => (Y) )
         => ( ( ? [Uu: $o] : ( Xa = vEBT_Leaf((Uu),$true) )
             => (Y) )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy) )
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) )
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1524_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat) )
     => ( ! [A3: $o,Uw: $o] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw)),aa_nat_nat(suc,zero_zero_nat)) )
       => ( ! [A3: $o,B3: $o,Va: nat] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va))) )
         => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3),Vb2) )
           => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve),Vf) )
             => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Vj) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_1525_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,B3: $o] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B3)),zero_zero_nat) )
     => ( ! [Uv: $o,Uw: $o,N: nat] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N)) )
       => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2),Va3) )
         => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd),Ve) )
           => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Vi) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_1526_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X: nat] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),X) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),X) )
         => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2),X) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_1527_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [Xa: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X: nat] : ( Xa != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),X) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),X) )
       => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2),X) )
         => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),X) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : ( Xa != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),X) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_1528_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] : ( Xa != vEBT_Leaf((A3),(B3)) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_1529_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)) ).

% List.finite_set
tff(fact_1530_List_Ofinite__set,axiom,
    ! [Xs: list_real] : aa_set_real_o(finite_finite_real,aa_lis3769085826154824284t_real(set_real2,Xs)) ).

% List.finite_set
tff(fact_1531_List_Ofinite__set,axiom,
    ! [Xs: list_o] : aa_set_o_o(finite_finite_o,aa_list_o_set_o(set_o2,Xs)) ).

% List.finite_set
tff(fact_1532_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,aa_list_nat_set_nat(set_nat2,Xs)) ).

% List.finite_set
tff(fact_1533_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,aa_list_int_set_int(set_int2,Xs)) ).

% List.finite_set
tff(fact_1534_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_1535_List_Ofinite__set,axiom,
    ! [Xs: list_Code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_Code_integer2(Xs)) ).

% List.finite_set
tff(fact_1536_delete__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa)) = minus_minus_set_nat(vEBT_set_vebt(T),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_1537_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ( Xa != vEBT_Leaf($false,$false) )
     => ( ! [Uv: $o] : ( Xa != vEBT_Leaf($true,(Uv)) )
       => ( ! [Uu: $o] : ( Xa != vEBT_Leaf((Uu),$true) )
         => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_1538_insertCI,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( ~ aa_set_o_o(member_o((A)),B4)
       => ( (A)
        <=> (B) ) )
     => aa_set_o_o(member_o((A)),aa_set_o_set_o(insert_o((B)),B4)) ) ).

% insertCI
tff(fact_1539_insertCI,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( ~ aa_set_nat_o(member_nat(A),B4)
       => ( A = B ) )
     => aa_set_nat_o(member_nat(A),aa_set_nat_set_nat(insert_nat(B),B4)) ) ).

% insertCI
tff(fact_1540_insertCI,axiom,
    ! [A: vEBT_VEBT,B4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B4)
       => ( A = B ) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),B4)) ) ).

% insertCI
tff(fact_1541_insertCI,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( ~ aa_set_real_o(member_real(A),B4)
       => ( A = B ) )
     => aa_set_real_o(member_real(A),aa_set_real_set_real(insert_real(B),B4)) ) ).

% insertCI
tff(fact_1542_insertCI,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( ( ~ aa_set_int_o(member_int(A),B4)
       => ( A = B ) )
     => aa_set_int_o(member_int(A),aa_set_int_set_int(insert_int(B),B4)) ) ).

% insertCI
tff(fact_1543_insertCI,axiom,
    ! [A: set_nat,B4: set_set_nat,B: set_nat] :
      ( ( ~ aa_set_set_nat_o(member_set_nat(A),B4)
       => ( A = B ) )
     => aa_set_set_nat_o(member_set_nat(A),aa_set1013257548857649334et_nat(insert_set_nat(B),B4)) ) ).

% insertCI
tff(fact_1544_insert__iff,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),aa_set_o_set_o(insert_o((B)),A4))
    <=> ( ( (A)
        <=> (B) )
        | aa_set_o_o(member_o((A)),A4) ) ) ).

% insert_iff
tff(fact_1545_insert__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),aa_set_nat_set_nat(insert_nat(B),A4))
    <=> ( ( A = B )
        | aa_set_nat_o(member_nat(A),A4) ) ) ).

% insert_iff
tff(fact_1546_insert__iff,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),A4))
    <=> ( ( A = B )
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4) ) ) ).

% insert_iff
tff(fact_1547_insert__iff,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),aa_set_real_set_real(insert_real(B),A4))
    <=> ( ( A = B )
        | aa_set_real_o(member_real(A),A4) ) ) ).

% insert_iff
tff(fact_1548_insert__iff,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),aa_set_int_set_int(insert_int(B),A4))
    <=> ( ( A = B )
        | aa_set_int_o(member_int(A),A4) ) ) ).

% insert_iff
tff(fact_1549_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),aa_set1013257548857649334et_nat(insert_set_nat(B),A4))
    <=> ( ( A = B )
        | aa_set_set_nat_o(member_set_nat(A),A4) ) ) ).

% insert_iff
tff(fact_1550_insert__absorb2,axiom,
    ! [Xa: nat,A4: set_nat] : ( aa_set_nat_set_nat(insert_nat(Xa),aa_set_nat_set_nat(insert_nat(Xa),A4)) = aa_set_nat_set_nat(insert_nat(Xa),A4) ) ).

% insert_absorb2
tff(fact_1551_insert__absorb2,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT] : ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4) ) ).

% insert_absorb2
tff(fact_1552_insert__absorb2,axiom,
    ! [Xa: int,A4: set_int] : ( aa_set_int_set_int(insert_int(Xa),aa_set_int_set_int(insert_int(Xa),A4)) = aa_set_int_set_int(insert_int(Xa),A4) ) ).

% insert_absorb2
tff(fact_1553_insert__absorb2,axiom,
    ! [Xa: $o,A4: set_o] : ( aa_set_o_set_o(insert_o((Xa)),aa_set_o_set_o(insert_o((Xa)),A4)) = aa_set_o_set_o(insert_o((Xa)),A4) ) ).

% insert_absorb2
tff(fact_1554_DiffI,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A4)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B4)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A4,B4)) ) ) ).

% DiffI
tff(fact_1555_DiffI,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C2),A4)
     => ( ~ aa_set_real_o(member_real(C2),B4)
       => aa_set_real_o(member_real(C2),minus_minus_set_real(A4,B4)) ) ) ).

% DiffI
tff(fact_1556_DiffI,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),A4)
     => ( ~ aa_set_int_o(member_int(C2),B4)
       => aa_set_int_o(member_int(C2),minus_minus_set_int(A4,B4)) ) ) ).

% DiffI
tff(fact_1557_DiffI,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),A4)
     => ( ~ aa_set_set_nat_o(member_set_nat(C2),B4)
       => aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4)) ) ) ).

% DiffI
tff(fact_1558_DiffI,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),A4)
     => ( ~ aa_set_nat_o(member_nat(C2),B4)
       => aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A4,B4)) ) ) ).

% DiffI
tff(fact_1559_Diff__iff,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A4,B4))
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A4)
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B4) ) ) ).

% Diff_iff
tff(fact_1560_Diff__iff,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A4,B4))
    <=> ( aa_set_real_o(member_real(C2),A4)
        & ~ aa_set_real_o(member_real(C2),B4) ) ) ).

% Diff_iff
tff(fact_1561_Diff__iff,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),minus_minus_set_int(A4,B4))
    <=> ( aa_set_int_o(member_int(C2),A4)
        & ~ aa_set_int_o(member_int(C2),B4) ) ) ).

% Diff_iff
tff(fact_1562_Diff__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4))
    <=> ( aa_set_set_nat_o(member_set_nat(C2),A4)
        & ~ aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% Diff_iff
tff(fact_1563_Diff__iff,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A4,B4))
    <=> ( aa_set_nat_o(member_nat(C2),A4)
        & ~ aa_set_nat_o(member_nat(C2),B4) ) ) ).

% Diff_iff
tff(fact_1564_Diff__idemp,axiom,
    ! [A4: set_nat,B4: set_nat] : ( minus_minus_set_nat(minus_minus_set_nat(A4,B4),B4) = minus_minus_set_nat(A4,B4) ) ).

% Diff_idemp
tff(fact_1565_set__vebt__delete,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( vEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa)) = minus_minus_set_nat(vEBT_set_vebt(T),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) ) ) ).

% set_vebt_delete
tff(fact_1566_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa)) = minus_minus_set_nat(vEBT_VEBT_set_vebt(T),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_1567_singletonI,axiom,
    ! [A: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)) ).

% singletonI
tff(fact_1568_singletonI,axiom,
    ! [A: real] : aa_set_real_o(member_real(A),aa_set_real_set_real(insert_real(A),bot_bot_set_real)) ).

% singletonI
tff(fact_1569_singletonI,axiom,
    ! [A: set_nat] : aa_set_set_nat_o(member_set_nat(A),aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat)) ).

% singletonI
tff(fact_1570_singletonI,axiom,
    ! [A: nat] : aa_set_nat_o(member_nat(A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)) ).

% singletonI
tff(fact_1571_singletonI,axiom,
    ! [A: int] : aa_set_int_o(member_int(A),aa_set_int_set_int(insert_int(A),bot_bot_set_int)) ).

% singletonI
tff(fact_1572_singletonI,axiom,
    ! [A: $o] : aa_set_o_o(member_o((A)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o)) ).

% singletonI
tff(fact_1573_finite__insert,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A4))
    <=> aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4) ) ).

% finite_insert
tff(fact_1574_finite__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(finite_finite_o,aa_set_o_set_o(insert_o((A)),A4))
    <=> aa_set_o_o(finite_finite_o,A4) ) ).

% finite_insert
tff(fact_1575_finite__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(insert_nat(A),A4))
    <=> aa_set_nat_o(finite_finite_nat,A4) ) ).

% finite_insert
tff(fact_1576_finite__insert,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(insert_int(A),A4))
    <=> aa_set_int_o(finite_finite_int,A4) ) ).

% finite_insert
tff(fact_1577_finite__insert,axiom,
    ! [A: complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A,A4))
    <=> aa_set_complex_o(finite3207457112153483333omplex,A4) ) ).

% finite_insert
tff(fact_1578_finite__insert,axiom,
    ! [A: code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,insert_Code_integer(A,A4))
    <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ).

% finite_insert
tff(fact_1579_insert__subset,axiom,
    ! [Xa: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(insert_o((Xa)),A4)),B4)
    <=> ( aa_set_o_o(member_o((Xa)),B4)
        & aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ) ).

% insert_subset
tff(fact_1580_insert__subset,axiom,
    ! [Xa: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(insert_nat(Xa),A4)),B4)
    <=> ( aa_set_nat_o(member_nat(Xa),B4)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ) ).

% insert_subset
tff(fact_1581_insert__subset,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)),B4)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B4)
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4) ) ) ).

% insert_subset
tff(fact_1582_insert__subset,axiom,
    ! [Xa: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(aa_set_real_set_real(insert_real(Xa),A4)),B4)
    <=> ( aa_set_real_o(member_real(Xa),B4)
        & aa_set_real_o(ord_less_eq_set_real(A4),B4) ) ) ).

% insert_subset
tff(fact_1583_insert__subset,axiom,
    ! [Xa: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4)),B4)
    <=> ( aa_set_set_nat_o(member_set_nat(Xa),B4)
        & aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4) ) ) ).

% insert_subset
tff(fact_1584_insert__subset,axiom,
    ! [Xa: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(insert_int(Xa),A4)),B4)
    <=> ( aa_set_int_o(member_int(Xa),B4)
        & aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ) ).

% insert_subset
tff(fact_1585_Diff__insert0,axiom,
    ! [Xa: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A4)
     => ( minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),B4)) = minus_minus_set_o(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1586_Diff__insert0,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),B4)) = minus_5127226145743854075T_VEBT(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1587_Diff__insert0,axiom,
    ! [Xa: real,A4: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A4)
     => ( minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),B4)) = minus_minus_set_real(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1588_Diff__insert0,axiom,
    ! [Xa: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A4)
     => ( minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),B4)) = minus_minus_set_int(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1589_Diff__insert0,axiom,
    ! [Xa: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A4)
     => ( minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),B4)) = minus_2163939370556025621et_nat(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1590_Diff__insert0,axiom,
    ! [Xa: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A4)
     => ( minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),B4)) = minus_minus_set_nat(A4,B4) ) ) ).

% Diff_insert0
tff(fact_1591_insert__Diff1,axiom,
    ! [Xa: $o,B4: set_o,A4: set_o] :
      ( aa_set_o_o(member_o((Xa)),B4)
     => ( minus_minus_set_o(aa_set_o_set_o(insert_o((Xa)),A4),B4) = minus_minus_set_o(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1592_insert__Diff1,axiom,
    ! [Xa: vEBT_VEBT,B4: set_VEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B4)
     => ( minus_5127226145743854075T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4),B4) = minus_5127226145743854075T_VEBT(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1593_insert__Diff1,axiom,
    ! [Xa: real,B4: set_real,A4: set_real] :
      ( aa_set_real_o(member_real(Xa),B4)
     => ( minus_minus_set_real(aa_set_real_set_real(insert_real(Xa),A4),B4) = minus_minus_set_real(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1594_insert__Diff1,axiom,
    ! [Xa: int,B4: set_int,A4: set_int] :
      ( aa_set_int_o(member_int(Xa),B4)
     => ( minus_minus_set_int(aa_set_int_set_int(insert_int(Xa),A4),B4) = minus_minus_set_int(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1595_insert__Diff1,axiom,
    ! [Xa: set_nat,B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),B4)
     => ( minus_2163939370556025621et_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4),B4) = minus_2163939370556025621et_nat(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1596_insert__Diff1,axiom,
    ! [Xa: nat,B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),B4)
     => ( minus_minus_set_nat(aa_set_nat_set_nat(insert_nat(Xa),A4),B4) = minus_minus_set_nat(A4,B4) ) ) ).

% insert_Diff1
tff(fact_1597_singleton__insert__inj__eq_H,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A4) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT) )
    <=> ( ( A = B )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1598_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A4: set_nat,B: nat] :
      ( ( aa_set_nat_set_nat(insert_nat(A),A4) = aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1599_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A4: set_o,B: $o] :
      ( ( aa_set_o_set_o(insert_o((A)),A4) = aa_set_o_set_o(insert_o((B)),bot_bot_set_o) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A4),aa_set_o_set_o(insert_o((B)),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1600_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A4: set_int,B: int] :
      ( ( aa_set_int_set_int(insert_int(A),A4) = aa_set_int_set_int(insert_int(B),bot_bot_set_int) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(insert_int(B),bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1601_singleton__insert__inj__eq,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A4) )
    <=> ( ( A = B )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1602_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A4: set_nat] :
      ( ( aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) = aa_set_nat_set_nat(insert_nat(A),A4) )
    <=> ( ( A = B )
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1603_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A4: set_o] :
      ( ( aa_set_o_set_o(insert_o((B)),bot_bot_set_o) = aa_set_o_set_o(insert_o((A)),A4) )
    <=> ( ( (A)
        <=> (B) )
        & aa_set_o_o(ord_less_eq_set_o(A4),aa_set_o_set_o(insert_o((B)),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1604_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A4: set_int] :
      ( ( aa_set_int_set_int(insert_int(B),bot_bot_set_int) = aa_set_int_set_int(insert_int(A),A4) )
    <=> ( ( A = B )
        & aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(insert_int(B),bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1605_insert__Diff__single,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] : ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A4) ) ).

% insert_Diff_single
tff(fact_1606_insert__Diff__single,axiom,
    ! [A: int,A4: set_int] : ( aa_set_int_set_int(insert_int(A),minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),bot_bot_set_int))) = aa_set_int_set_int(insert_int(A),A4) ) ).

% insert_Diff_single
tff(fact_1607_insert__Diff__single,axiom,
    ! [A: $o,A4: set_o] : ( aa_set_o_set_o(insert_o((A)),minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),bot_bot_set_o))) = aa_set_o_set_o(insert_o((A)),A4) ) ).

% insert_Diff_single
tff(fact_1608_insert__Diff__single,axiom,
    ! [A: nat,A4: set_nat] : ( aa_set_nat_set_nat(insert_nat(A),minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = aa_set_nat_set_nat(insert_nat(A),A4) ) ).

% insert_Diff_single
tff(fact_1609_finite__Diff__insert,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4)))
    <=> aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A4,B4)) ) ).

% finite_Diff_insert
tff(fact_1610_finite__Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),B4)))
    <=> aa_set_o_o(finite_finite_o,minus_minus_set_o(A4,B4)) ) ).

% finite_Diff_insert
tff(fact_1611_finite__Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),B4)))
    <=> aa_set_int_o(finite_finite_int,minus_minus_set_int(A4,B4)) ) ).

% finite_Diff_insert
tff(fact_1612_finite__Diff__insert,axiom,
    ! [A4: set_complex,A: complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,insert_complex(A,B4)))
    <=> aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A4,B4)) ) ).

% finite_Diff_insert
tff(fact_1613_finite__Diff__insert,axiom,
    ! [A4: set_Code_integer,A: code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A4,insert_Code_integer(A,B4)))
    <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A4,B4)) ) ).

% finite_Diff_insert
tff(fact_1614_finite__Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(A),B4)))
    <=> aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A4,B4)) ) ).

% finite_Diff_insert
tff(fact_1615_DiffE,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A4,B4))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A4)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B4) ) ) ).

% DiffE
tff(fact_1616_DiffE,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A4,B4))
     => ~ ( aa_set_real_o(member_real(C2),A4)
         => aa_set_real_o(member_real(C2),B4) ) ) ).

% DiffE
tff(fact_1617_DiffE,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),minus_minus_set_int(A4,B4))
     => ~ ( aa_set_int_o(member_int(C2),A4)
         => aa_set_int_o(member_int(C2),B4) ) ) ).

% DiffE
tff(fact_1618_DiffE,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4))
     => ~ ( aa_set_set_nat_o(member_set_nat(C2),A4)
         => aa_set_set_nat_o(member_set_nat(C2),B4) ) ) ).

% DiffE
tff(fact_1619_DiffE,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A4,B4))
     => ~ ( aa_set_nat_o(member_nat(C2),A4)
         => aa_set_nat_o(member_nat(C2),B4) ) ) ).

% DiffE
tff(fact_1620_DiffD1,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A4,B4))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A4) ) ).

% DiffD1
tff(fact_1621_DiffD1,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A4,B4))
     => aa_set_real_o(member_real(C2),A4) ) ).

% DiffD1
tff(fact_1622_DiffD1,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),minus_minus_set_int(A4,B4))
     => aa_set_int_o(member_int(C2),A4) ) ).

% DiffD1
tff(fact_1623_DiffD1,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4))
     => aa_set_set_nat_o(member_set_nat(C2),A4) ) ).

% DiffD1
tff(fact_1624_DiffD1,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A4,B4))
     => aa_set_nat_o(member_nat(C2),A4) ) ).

% DiffD1
tff(fact_1625_DiffD2,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A4,B4))
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B4) ) ).

% DiffD2
tff(fact_1626_DiffD2,axiom,
    ! [C2: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A4,B4))
     => ~ aa_set_real_o(member_real(C2),B4) ) ).

% DiffD2
tff(fact_1627_DiffD2,axiom,
    ! [C2: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(C2),minus_minus_set_int(A4,B4))
     => ~ aa_set_int_o(member_int(C2),B4) ) ).

% DiffD2
tff(fact_1628_DiffD2,axiom,
    ! [C2: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),minus_2163939370556025621et_nat(A4,B4))
     => ~ aa_set_set_nat_o(member_set_nat(C2),B4) ) ).

% DiffD2
tff(fact_1629_DiffD2,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A4,B4))
     => ~ aa_set_nat_o(member_nat(C2),B4) ) ).

% DiffD2
tff(fact_1630_insert__Diff__if,axiom,
    ! [Xa: $o,A4: set_o,B4: set_o] :
      ( minus_minus_set_o(aa_set_o_set_o(insert_o((Xa)),A4),B4) = $ite(aa_set_o_o(member_o((Xa)),B4),minus_minus_set_o(A4,B4),aa_set_o_set_o(insert_o((Xa)),minus_minus_set_o(A4,B4))) ) ).

% insert_Diff_if
tff(fact_1631_insert__Diff__if,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( minus_5127226145743854075T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4),B4) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B4),minus_5127226145743854075T_VEBT(A4,B4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),minus_5127226145743854075T_VEBT(A4,B4))) ) ).

% insert_Diff_if
tff(fact_1632_insert__Diff__if,axiom,
    ! [Xa: real,A4: set_real,B4: set_real] :
      ( minus_minus_set_real(aa_set_real_set_real(insert_real(Xa),A4),B4) = $ite(aa_set_real_o(member_real(Xa),B4),minus_minus_set_real(A4,B4),aa_set_real_set_real(insert_real(Xa),minus_minus_set_real(A4,B4))) ) ).

% insert_Diff_if
tff(fact_1633_insert__Diff__if,axiom,
    ! [Xa: int,A4: set_int,B4: set_int] :
      ( minus_minus_set_int(aa_set_int_set_int(insert_int(Xa),A4),B4) = $ite(aa_set_int_o(member_int(Xa),B4),minus_minus_set_int(A4,B4),aa_set_int_set_int(insert_int(Xa),minus_minus_set_int(A4,B4))) ) ).

% insert_Diff_if
tff(fact_1634_insert__Diff__if,axiom,
    ! [Xa: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( minus_2163939370556025621et_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4),B4) = $ite(aa_set_set_nat_o(member_set_nat(Xa),B4),minus_2163939370556025621et_nat(A4,B4),aa_set1013257548857649334et_nat(insert_set_nat(Xa),minus_2163939370556025621et_nat(A4,B4))) ) ).

% insert_Diff_if
tff(fact_1635_insert__Diff__if,axiom,
    ! [Xa: nat,A4: set_nat,B4: set_nat] :
      ( minus_minus_set_nat(aa_set_nat_set_nat(insert_nat(Xa),A4),B4) = $ite(aa_set_nat_o(member_nat(Xa),B4),minus_minus_set_nat(A4,B4),aa_set_nat_set_nat(insert_nat(Xa),minus_minus_set_nat(A4,B4))) ) ).

% insert_Diff_if
tff(fact_1636_insertE,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),aa_set_o_set_o(insert_o((B)),A4))
     => ( ( (A)
        <=> ~ (B) )
       => aa_set_o_o(member_o((A)),A4) ) ) ).

% insertE
tff(fact_1637_insertE,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),aa_set_nat_set_nat(insert_nat(B),A4))
     => ( ( A != B )
       => aa_set_nat_o(member_nat(A),A4) ) ) ).

% insertE
tff(fact_1638_insertE,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),A4))
     => ( ( A != B )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4) ) ) ).

% insertE
tff(fact_1639_insertE,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),aa_set_real_set_real(insert_real(B),A4))
     => ( ( A != B )
       => aa_set_real_o(member_real(A),A4) ) ) ).

% insertE
tff(fact_1640_insertE,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),aa_set_int_set_int(insert_int(B),A4))
     => ( ( A != B )
       => aa_set_int_o(member_int(A),A4) ) ) ).

% insertE
tff(fact_1641_insertE,axiom,
    ! [A: set_nat,B: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),aa_set1013257548857649334et_nat(insert_set_nat(B),A4))
     => ( ( A != B )
       => aa_set_set_nat_o(member_set_nat(A),A4) ) ) ).

% insertE
tff(fact_1642_insertI1,axiom,
    ! [A: $o,B4: set_o] : aa_set_o_o(member_o((A)),aa_set_o_set_o(insert_o((A)),B4)) ).

% insertI1
tff(fact_1643_insertI1,axiom,
    ! [A: nat,B4: set_nat] : aa_set_nat_o(member_nat(A),aa_set_nat_set_nat(insert_nat(A),B4)) ).

% insertI1
tff(fact_1644_insertI1,axiom,
    ! [A: vEBT_VEBT,B4: set_VEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4)) ).

% insertI1
tff(fact_1645_insertI1,axiom,
    ! [A: real,B4: set_real] : aa_set_real_o(member_real(A),aa_set_real_set_real(insert_real(A),B4)) ).

% insertI1
tff(fact_1646_insertI1,axiom,
    ! [A: int,B4: set_int] : aa_set_int_o(member_int(A),aa_set_int_set_int(insert_int(A),B4)) ).

% insertI1
tff(fact_1647_insertI1,axiom,
    ! [A: set_nat,B4: set_set_nat] : aa_set_set_nat_o(member_set_nat(A),aa_set1013257548857649334et_nat(insert_set_nat(A),B4)) ).

% insertI1
tff(fact_1648_insertI2,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( aa_set_o_o(member_o((A)),B4)
     => aa_set_o_o(member_o((A)),aa_set_o_set_o(insert_o((B)),B4)) ) ).

% insertI2
tff(fact_1649_insertI2,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(member_nat(A),B4)
     => aa_set_nat_o(member_nat(A),aa_set_nat_set_nat(insert_nat(B),B4)) ) ).

% insertI2
tff(fact_1650_insertI2,axiom,
    ! [A: vEBT_VEBT,B4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B4)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),B4)) ) ).

% insertI2
tff(fact_1651_insertI2,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( aa_set_real_o(member_real(A),B4)
     => aa_set_real_o(member_real(A),aa_set_real_set_real(insert_real(B),B4)) ) ).

% insertI2
tff(fact_1652_insertI2,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( aa_set_int_o(member_int(A),B4)
     => aa_set_int_o(member_int(A),aa_set_int_set_int(insert_int(B),B4)) ) ).

% insertI2
tff(fact_1653_insertI2,axiom,
    ! [A: set_nat,B4: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),B4)
     => aa_set_set_nat_o(member_set_nat(A),aa_set1013257548857649334et_nat(insert_set_nat(B),B4)) ) ).

% insertI2
tff(fact_1654_Set_Oset__insert,axiom,
    ! [Xa: $o,A4: set_o] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ~ ! [B7: set_o] :
            ( ( A4 = aa_set_o_set_o(insert_o((Xa)),B7) )
           => aa_set_o_o(member_o((Xa)),B7) ) ) ).

% Set.set_insert
tff(fact_1655_Set_Oset__insert,axiom,
    ! [Xa: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ~ ! [B7: set_nat] :
            ( ( A4 = aa_set_nat_set_nat(insert_nat(Xa),B7) )
           => aa_set_nat_o(member_nat(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_1656_Set_Oset__insert,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ~ ! [B7: set_VEBT_VEBT] :
            ( ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),B7) )
           => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_1657_Set_Oset__insert,axiom,
    ! [Xa: real,A4: set_real] :
      ( aa_set_real_o(member_real(Xa),A4)
     => ~ ! [B7: set_real] :
            ( ( A4 = aa_set_real_set_real(insert_real(Xa),B7) )
           => aa_set_real_o(member_real(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_1658_Set_Oset__insert,axiom,
    ! [Xa: int,A4: set_int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ~ ! [B7: set_int] :
            ( ( A4 = aa_set_int_set_int(insert_int(Xa),B7) )
           => aa_set_int_o(member_int(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_1659_Set_Oset__insert,axiom,
    ! [Xa: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),A4)
     => ~ ! [B7: set_set_nat] :
            ( ( A4 = aa_set1013257548857649334et_nat(insert_set_nat(Xa),B7) )
           => aa_set_set_nat_o(member_set_nat(Xa),B7) ) ) ).

% Set.set_insert
tff(fact_1660_insert__ident,axiom,
    ! [Xa: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A4)
     => ( ~ aa_set_o_o(member_o((Xa)),B4)
       => ( ( aa_set_o_set_o(insert_o((Xa)),A4) = aa_set_o_set_o(insert_o((Xa)),B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_1661_insert__ident,axiom,
    ! [Xa: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A4)
     => ( ~ aa_set_nat_o(member_nat(Xa),B4)
       => ( ( aa_set_nat_set_nat(insert_nat(Xa),A4) = aa_set_nat_set_nat(insert_nat(Xa),B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_1662_insert__ident,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B4)
       => ( ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_1663_insert__ident,axiom,
    ! [Xa: real,A4: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A4)
     => ( ~ aa_set_real_o(member_real(Xa),B4)
       => ( ( aa_set_real_set_real(insert_real(Xa),A4) = aa_set_real_set_real(insert_real(Xa),B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_1664_insert__ident,axiom,
    ! [Xa: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A4)
     => ( ~ aa_set_int_o(member_int(Xa),B4)
       => ( ( aa_set_int_set_int(insert_int(Xa),A4) = aa_set_int_set_int(insert_int(Xa),B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_1665_insert__ident,axiom,
    ! [Xa: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A4)
     => ( ~ aa_set_set_nat_o(member_set_nat(Xa),B4)
       => ( ( aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4) = aa_set1013257548857649334et_nat(insert_set_nat(Xa),B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% insert_ident
tff(fact_1666_insert__absorb,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( aa_set_o_set_o(insert_o((A)),A4) = A4 ) ) ).

% insert_absorb
tff(fact_1667_insert__absorb,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_nat_set_nat(insert_nat(A),A4) = A4 ) ) ).

% insert_absorb
tff(fact_1668_insert__absorb,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4)
     => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A4) = A4 ) ) ).

% insert_absorb
tff(fact_1669_insert__absorb,axiom,
    ! [A: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),A4)
     => ( aa_set_real_set_real(insert_real(A),A4) = A4 ) ) ).

% insert_absorb
tff(fact_1670_insert__absorb,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( aa_set_int_set_int(insert_int(A),A4) = A4 ) ) ).

% insert_absorb
tff(fact_1671_insert__absorb,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A4)
     => ( aa_set1013257548857649334et_nat(insert_set_nat(A),A4) = A4 ) ) ).

% insert_absorb
tff(fact_1672_insert__eq__iff,axiom,
    ! [A: $o,A4: set_o,B: $o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((A)),A4)
     => ( ~ aa_set_o_o(member_o((B)),B4)
       => ( ( aa_set_o_set_o(insert_o((A)),A4) = aa_set_o_set_o(insert_o((B)),B4) )
        <=> $ite(
              (A) = (B),
              A4 = B4,
              ? [C4: set_o] :
                ( ( A4 = aa_set_o_set_o(insert_o((B)),C4) )
                & ~ aa_set_o_o(member_o((B)),C4)
                & ( B4 = aa_set_o_set_o(insert_o((A)),C4) )
                & ~ aa_set_o_o(member_o((A)),C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1673_insert__eq__iff,axiom,
    ! [A: nat,A4: set_nat,B: nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A),A4)
     => ( ~ aa_set_nat_o(member_nat(B),B4)
       => ( ( aa_set_nat_set_nat(insert_nat(A),A4) = aa_set_nat_set_nat(insert_nat(B),B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C4: set_nat] :
                ( ( A4 = aa_set_nat_set_nat(insert_nat(B),C4) )
                & ~ aa_set_nat_o(member_nat(B),C4)
                & ( B4 = aa_set_nat_set_nat(insert_nat(A),C4) )
                & ~ aa_set_nat_o(member_nat(A),C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1674_insert__eq__iff,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),B4)
       => ( ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A4) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C4: set_VEBT_VEBT] :
                ( ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),C4) )
                & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),C4)
                & ( B4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),C4) )
                & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1675_insert__eq__iff,axiom,
    ! [A: real,A4: set_real,B: real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(A),A4)
     => ( ~ aa_set_real_o(member_real(B),B4)
       => ( ( aa_set_real_set_real(insert_real(A),A4) = aa_set_real_set_real(insert_real(B),B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C4: set_real] :
                ( ( A4 = aa_set_real_set_real(insert_real(B),C4) )
                & ~ aa_set_real_o(member_real(B),C4)
                & ( B4 = aa_set_real_set_real(insert_real(A),C4) )
                & ~ aa_set_real_o(member_real(A),C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1676_insert__eq__iff,axiom,
    ! [A: int,A4: set_int,B: int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(A),A4)
     => ( ~ aa_set_int_o(member_int(B),B4)
       => ( ( aa_set_int_set_int(insert_int(A),A4) = aa_set_int_set_int(insert_int(B),B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C4: set_int] :
                ( ( A4 = aa_set_int_set_int(insert_int(B),C4) )
                & ~ aa_set_int_o(member_int(B),C4)
                & ( B4 = aa_set_int_set_int(insert_int(A),C4) )
                & ~ aa_set_int_o(member_int(A),C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1677_insert__eq__iff,axiom,
    ! [A: set_nat,A4: set_set_nat,B: set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(A),A4)
     => ( ~ aa_set_set_nat_o(member_set_nat(B),B4)
       => ( ( aa_set1013257548857649334et_nat(insert_set_nat(A),A4) = aa_set1013257548857649334et_nat(insert_set_nat(B),B4) )
        <=> $ite(
              A = B,
              A4 = B4,
              ? [C4: set_set_nat] :
                ( ( A4 = aa_set1013257548857649334et_nat(insert_set_nat(B),C4) )
                & ~ aa_set_set_nat_o(member_set_nat(B),C4)
                & ( B4 = aa_set1013257548857649334et_nat(insert_set_nat(A),C4) )
                & ~ aa_set_set_nat_o(member_set_nat(A),C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1678_insert__commute,axiom,
    ! [Xa: nat,Y: nat,A4: set_nat] : ( aa_set_nat_set_nat(insert_nat(Xa),aa_set_nat_set_nat(insert_nat(Y),A4)) = aa_set_nat_set_nat(insert_nat(Y),aa_set_nat_set_nat(insert_nat(Xa),A4)) ) ).

% insert_commute
tff(fact_1679_insert__commute,axiom,
    ! [Xa: vEBT_VEBT,Y: vEBT_VEBT,A4: set_VEBT_VEBT] : ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Y),A4)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Y),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) ) ).

% insert_commute
tff(fact_1680_insert__commute,axiom,
    ! [Xa: int,Y: int,A4: set_int] : ( aa_set_int_set_int(insert_int(Xa),aa_set_int_set_int(insert_int(Y),A4)) = aa_set_int_set_int(insert_int(Y),aa_set_int_set_int(insert_int(Xa),A4)) ) ).

% insert_commute
tff(fact_1681_insert__commute,axiom,
    ! [Xa: $o,Y: $o,A4: set_o] : ( aa_set_o_set_o(insert_o((Xa)),aa_set_o_set_o(insert_o((Y)),A4)) = aa_set_o_set_o(insert_o((Y)),aa_set_o_set_o(insert_o((Xa)),A4)) ) ).

% insert_commute
tff(fact_1682_mk__disjoint__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ? [B7: set_o] :
          ( ( A4 = aa_set_o_set_o(insert_o((A)),B7) )
          & ~ aa_set_o_o(member_o((A)),B7) ) ) ).

% mk_disjoint_insert
tff(fact_1683_mk__disjoint__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ? [B7: set_nat] :
          ( ( A4 = aa_set_nat_set_nat(insert_nat(A),B7) )
          & ~ aa_set_nat_o(member_nat(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_1684_mk__disjoint__insert,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4)
     => ? [B7: set_VEBT_VEBT] :
          ( ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B7) )
          & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_1685_mk__disjoint__insert,axiom,
    ! [A: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),A4)
     => ? [B7: set_real] :
          ( ( A4 = aa_set_real_set_real(insert_real(A),B7) )
          & ~ aa_set_real_o(member_real(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_1686_mk__disjoint__insert,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ? [B7: set_int] :
          ( ( A4 = aa_set_int_set_int(insert_int(A),B7) )
          & ~ aa_set_int_o(member_int(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_1687_mk__disjoint__insert,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A4)
     => ? [B7: set_set_nat] :
          ( ( A4 = aa_set1013257548857649334et_nat(insert_set_nat(A),B7) )
          & ~ aa_set_set_nat_o(member_set_nat(A),B7) ) ) ).

% mk_disjoint_insert
tff(fact_1688_singleton__inject,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT] :
      ( ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_1689_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) = aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_1690_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( aa_set_int_set_int(insert_int(A),bot_bot_set_int) = aa_set_int_set_int(insert_int(B),bot_bot_set_int) )
     => ( A = B ) ) ).

% singleton_inject
tff(fact_1691_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( aa_set_o_set_o(insert_o((A)),bot_bot_set_o) = aa_set_o_set_o(insert_o((B)),bot_bot_set_o) )
     => ( (A)
      <=> (B) ) ) ).

% singleton_inject
tff(fact_1692_insert__not__empty,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] : ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A4) != bot_bo8194388402131092736T_VEBT ) ).

% insert_not_empty
tff(fact_1693_insert__not__empty,axiom,
    ! [A: nat,A4: set_nat] : ( aa_set_nat_set_nat(insert_nat(A),A4) != bot_bot_set_nat ) ).

% insert_not_empty
tff(fact_1694_insert__not__empty,axiom,
    ! [A: int,A4: set_int] : ( aa_set_int_set_int(insert_int(A),A4) != bot_bot_set_int ) ).

% insert_not_empty
tff(fact_1695_insert__not__empty,axiom,
    ! [A: $o,A4: set_o] : ( aa_set_o_set_o(insert_o((A)),A4) != bot_bot_set_o ) ).

% insert_not_empty
tff(fact_1696_doubleton__eq__iff,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,C2: vEBT_VEBT,D: vEBT_VEBT] :
      ( ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C2),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(D),bot_bo8194388402131092736T_VEBT)) )
    <=> ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1697_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)) = aa_set_nat_set_nat(insert_nat(C2),aa_set_nat_set_nat(insert_nat(D),bot_bot_set_nat)) )
    <=> ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1698_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( aa_set_int_set_int(insert_int(A),aa_set_int_set_int(insert_int(B),bot_bot_set_int)) = aa_set_int_set_int(insert_int(C2),aa_set_int_set_int(insert_int(D),bot_bot_set_int)) )
    <=> ( ( ( A = C2 )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1699_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C2: $o,D: $o] :
      ( ( aa_set_o_set_o(insert_o((A)),aa_set_o_set_o(insert_o((B)),bot_bot_set_o)) = aa_set_o_set_o(insert_o((C2)),aa_set_o_set_o(insert_o((D)),bot_bot_set_o)) )
    <=> ( ( ( (A)
          <=> (C2) )
          & ( (B)
          <=> (D) ) )
        | ( ( (A)
          <=> (D) )
          & ( (B)
          <=> (C2) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1700_singleton__iff,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_1701_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real(B),aa_set_real_set_real(insert_real(A),bot_bot_set_real))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_1702_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_1703_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_1704_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),aa_set_int_set_int(insert_int(A),bot_bot_set_int))
    <=> ( B = A ) ) ).

% singleton_iff
tff(fact_1705_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o))
    <=> ( (B)
      <=> (A) ) ) ).

% singleton_iff
tff(fact_1706_singletonD,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))
     => ( B = A ) ) ).

% singletonD
tff(fact_1707_singletonD,axiom,
    ! [B: real,A: real] :
      ( aa_set_real_o(member_real(B),aa_set_real_set_real(insert_real(A),bot_bot_set_real))
     => ( B = A ) ) ).

% singletonD
tff(fact_1708_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(B),aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_1709_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))
     => ( B = A ) ) ).

% singletonD
tff(fact_1710_singletonD,axiom,
    ! [B: int,A: int] :
      ( aa_set_int_o(member_int(B),aa_set_int_set_int(insert_int(A),bot_bot_set_int))
     => ( B = A ) ) ).

% singletonD
tff(fact_1711_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( aa_set_o_o(member_o((B)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o))
     => ( (B)
      <=> (A) ) ) ).

% singletonD
tff(fact_1712_insert__minus__eq,axiom,
    ! [Xa: vEBT_VEBT,Y: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( Xa != Y )
     => ( minus_5127226145743854075T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Y),bot_bo8194388402131092736T_VEBT)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Y),bot_bo8194388402131092736T_VEBT))) ) ) ).

% insert_minus_eq
tff(fact_1713_insert__minus__eq,axiom,
    ! [Xa: int,Y: int,A4: set_int] :
      ( ( Xa != Y )
     => ( minus_minus_set_int(aa_set_int_set_int(insert_int(Xa),A4),aa_set_int_set_int(insert_int(Y),bot_bot_set_int)) = aa_set_int_set_int(insert_int(Xa),minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Y),bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_1714_insert__minus__eq,axiom,
    ! [Xa: $o,Y: $o,A4: set_o] :
      ( ~ ( (Xa)
        <=> (Y) )
     => ( minus_minus_set_o(aa_set_o_set_o(insert_o((Xa)),A4),aa_set_o_set_o(insert_o((Y)),bot_bot_set_o)) = aa_set_o_set_o(insert_o((Xa)),minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Y)),bot_bot_set_o))) ) ) ).

% insert_minus_eq
tff(fact_1715_insert__minus__eq,axiom,
    ! [Xa: nat,Y: nat,A4: set_nat] :
      ( ( Xa != Y )
     => ( minus_minus_set_nat(aa_set_nat_set_nat(insert_nat(Xa),A4),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)) = aa_set_nat_set_nat(insert_nat(Xa),minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_1716_set__minus__singleton__eq,axiom,
    ! [Xa: vEBT_VEBT,X4: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),X4)
     => ( minus_5127226145743854075T_VEBT(X4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)) = X4 ) ) ).

% set_minus_singleton_eq
tff(fact_1717_set__minus__singleton__eq,axiom,
    ! [Xa: real,X4: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),X4)
     => ( minus_minus_set_real(X4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)) = X4 ) ) ).

% set_minus_singleton_eq
tff(fact_1718_set__minus__singleton__eq,axiom,
    ! [Xa: set_nat,X4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),X4)
     => ( minus_2163939370556025621et_nat(X4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)) = X4 ) ) ).

% set_minus_singleton_eq
tff(fact_1719_set__minus__singleton__eq,axiom,
    ! [Xa: int,X4: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),X4)
     => ( minus_minus_set_int(X4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) = X4 ) ) ).

% set_minus_singleton_eq
tff(fact_1720_set__minus__singleton__eq,axiom,
    ! [Xa: $o,X4: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),X4)
     => ( minus_minus_set_o(X4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)) = X4 ) ) ).

% set_minus_singleton_eq
tff(fact_1721_set__minus__singleton__eq,axiom,
    ! [Xa: nat,X4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),X4)
     => ( minus_minus_set_nat(X4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) = X4 ) ) ).

% set_minus_singleton_eq
tff(fact_1722_Diff__insert__absorb,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( minus_5127226145743854075T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1723_Diff__insert__absorb,axiom,
    ! [Xa: real,A4: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A4)
     => ( minus_minus_set_real(aa_set_real_set_real(insert_real(Xa),A4),aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1724_Diff__insert__absorb,axiom,
    ! [Xa: set_nat,A4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A4)
     => ( minus_2163939370556025621et_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4),aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1725_Diff__insert__absorb,axiom,
    ! [Xa: int,A4: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A4)
     => ( minus_minus_set_int(aa_set_int_set_int(insert_int(Xa),A4),aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1726_Diff__insert__absorb,axiom,
    ! [Xa: $o,A4: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A4)
     => ( minus_minus_set_o(aa_set_o_set_o(insert_o((Xa)),A4),aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1727_Diff__insert__absorb,axiom,
    ! [Xa: nat,A4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A4)
     => ( minus_minus_set_nat(aa_set_nat_set_nat(insert_nat(Xa),A4),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_1728_Diff__insert2,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] : ( minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)),B4) ) ).

% Diff_insert2
tff(fact_1729_Diff__insert2,axiom,
    ! [A4: set_int,A: int,B4: set_int] : ( minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),B4)) = minus_minus_set_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),bot_bot_set_int)),B4) ) ).

% Diff_insert2
tff(fact_1730_Diff__insert2,axiom,
    ! [A4: set_o,A: $o,B4: set_o] : ( minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),B4)) = minus_minus_set_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),bot_bot_set_o)),B4) ) ).

% Diff_insert2
tff(fact_1731_Diff__insert2,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : ( minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(A),B4)) = minus_minus_set_nat(minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)),B4) ) ).

% Diff_insert2
tff(fact_1732_insert__Diff,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4)
     => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))) = A4 ) ) ).

% insert_Diff
tff(fact_1733_insert__Diff,axiom,
    ! [A: real,A4: set_real] :
      ( aa_set_real_o(member_real(A),A4)
     => ( aa_set_real_set_real(insert_real(A),minus_minus_set_real(A4,aa_set_real_set_real(insert_real(A),bot_bot_set_real))) = A4 ) ) ).

% insert_Diff
tff(fact_1734_insert__Diff,axiom,
    ! [A: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A4)
     => ( aa_set1013257548857649334et_nat(insert_set_nat(A),minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_1735_insert__Diff,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( aa_set_int_set_int(insert_int(A),minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),bot_bot_set_int))) = A4 ) ) ).

% insert_Diff
tff(fact_1736_insert__Diff,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( aa_set_o_set_o(insert_o((A)),minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),bot_bot_set_o))) = A4 ) ) ).

% insert_Diff
tff(fact_1737_insert__Diff,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_nat_set_nat(insert_nat(A),minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = A4 ) ) ).

% insert_Diff
tff(fact_1738_Diff__insert,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] : ( minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A4,B4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)) ) ).

% Diff_insert
tff(fact_1739_Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] : ( minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),B4)) = minus_minus_set_int(minus_minus_set_int(A4,B4),aa_set_int_set_int(insert_int(A),bot_bot_set_int)) ) ).

% Diff_insert
tff(fact_1740_Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] : ( minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),B4)) = minus_minus_set_o(minus_minus_set_o(A4,B4),aa_set_o_set_o(insert_o((A)),bot_bot_set_o)) ) ).

% Diff_insert
tff(fact_1741_Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] : ( minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(A),B4)) = minus_minus_set_nat(minus_minus_set_nat(A4,B4),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)) ) ).

% Diff_insert
tff(fact_1742_finite_OinsertI,axiom,
    ! [A4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),A4)) ) ).

% finite.insertI
tff(fact_1743_finite_OinsertI,axiom,
    ! [A4: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => aa_set_o_o(finite_finite_o,aa_set_o_set_o(insert_o((A)),A4)) ) ).

% finite.insertI
tff(fact_1744_finite_OinsertI,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(insert_nat(A),A4)) ) ).

% finite.insertI
tff(fact_1745_finite_OinsertI,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_set_int_o(finite_finite_int,aa_set_int_set_int(insert_int(A),A4)) ) ).

% finite.insertI
tff(fact_1746_finite_OinsertI,axiom,
    ! [A4: set_complex,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A,A4)) ) ).

% finite.insertI
tff(fact_1747_finite_OinsertI,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,insert_Code_integer(A,A4)) ) ).

% finite.insertI
tff(fact_1748_subset__insertI2,axiom,
    ! [A4: set_nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(insert_nat(B),B4)) ) ).

% subset_insertI2
tff(fact_1749_subset__insertI2,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),B4)) ) ).

% subset_insertI2
tff(fact_1750_subset__insertI2,axiom,
    ! [A4: set_o,B4: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => aa_set_o_o(ord_less_eq_set_o(A4),aa_set_o_set_o(insert_o((B)),B4)) ) ).

% subset_insertI2
tff(fact_1751_subset__insertI2,axiom,
    ! [A4: set_int,B4: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(insert_int(B),B4)) ) ).

% subset_insertI2
tff(fact_1752_subset__insertI,axiom,
    ! [B4: set_nat,A: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(insert_nat(A),B4)) ).

% subset_insertI
tff(fact_1753_subset__insertI,axiom,
    ! [B4: set_VEBT_VEBT,A: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4)) ).

% subset_insertI
tff(fact_1754_subset__insertI,axiom,
    ! [B4: set_o,A: $o] : aa_set_o_o(ord_less_eq_set_o(B4),aa_set_o_set_o(insert_o((A)),B4)) ).

% subset_insertI
tff(fact_1755_subset__insertI,axiom,
    ! [B4: set_int,A: int] : aa_set_int_o(ord_less_eq_set_int(B4),aa_set_int_set_int(insert_int(A),B4)) ).

% subset_insertI
tff(fact_1756_subset__insert,axiom,
    ! [Xa: $o,A4: set_o,B4: set_o] :
      ( ~ aa_set_o_o(member_o((Xa)),A4)
     => ( aa_set_o_o(ord_less_eq_set_o(A4),aa_set_o_set_o(insert_o((Xa)),B4))
      <=> aa_set_o_o(ord_less_eq_set_o(A4),B4) ) ) ).

% subset_insert
tff(fact_1757_subset__insert,axiom,
    ! [Xa: nat,A4: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(insert_nat(Xa),B4))
      <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4) ) ) ).

% subset_insert
tff(fact_1758_subset__insert,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),B4))
      <=> aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4) ) ) ).

% subset_insert
tff(fact_1759_subset__insert,axiom,
    ! [Xa: real,A4: set_real,B4: set_real] :
      ( ~ aa_set_real_o(member_real(Xa),A4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),aa_set_real_set_real(insert_real(Xa),B4))
      <=> aa_set_real_o(ord_less_eq_set_real(A4),B4) ) ) ).

% subset_insert
tff(fact_1760_subset__insert,axiom,
    ! [Xa: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),A4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),aa_set1013257548857649334et_nat(insert_set_nat(Xa),B4))
      <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4) ) ) ).

% subset_insert
tff(fact_1761_subset__insert,axiom,
    ! [Xa: int,A4: set_int,B4: set_int] :
      ( ~ aa_set_int_o(member_int(Xa),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(insert_int(Xa),B4))
      <=> aa_set_int_o(ord_less_eq_set_int(A4),B4) ) ) ).

% subset_insert
tff(fact_1762_insert__mono,axiom,
    ! [C3: set_nat,D4: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C3),D4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(insert_nat(A),C3)),aa_set_nat_set_nat(insert_nat(A),D4)) ) ).

% insert_mono
tff(fact_1763_insert__mono,axiom,
    ! [C3: set_VEBT_VEBT,D4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(C3),D4)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),C3)),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),D4)) ) ).

% insert_mono
tff(fact_1764_insert__mono,axiom,
    ! [C3: set_o,D4: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(C3),D4)
     => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(insert_o((A)),C3)),aa_set_o_set_o(insert_o((A)),D4)) ) ).

% insert_mono
tff(fact_1765_insert__mono,axiom,
    ! [C3: set_int,D4: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(C3),D4)
     => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(insert_int(A),C3)),aa_set_int_set_int(insert_int(A),D4)) ) ).

% insert_mono
tff(fact_1766_insert__subsetI,axiom,
    ! [Xa: $o,A4: set_o,X4: set_o] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( aa_set_o_o(ord_less_eq_set_o(X4),A4)
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(insert_o((Xa)),X4)),A4) ) ) ).

% insert_subsetI
tff(fact_1767_insert__subsetI,axiom,
    ! [Xa: nat,A4: set_nat,X4: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X4),A4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(insert_nat(Xa),X4)),A4) ) ) ).

% insert_subsetI
tff(fact_1768_insert__subsetI,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,X4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(X4),A4)
       => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),X4)),A4) ) ) ).

% insert_subsetI
tff(fact_1769_insert__subsetI,axiom,
    ! [Xa: real,A4: set_real,X4: set_real] :
      ( aa_set_real_o(member_real(Xa),A4)
     => ( aa_set_real_o(ord_less_eq_set_real(X4),A4)
       => aa_set_real_o(ord_less_eq_set_real(aa_set_real_set_real(insert_real(Xa),X4)),A4) ) ) ).

% insert_subsetI
tff(fact_1770_insert__subsetI,axiom,
    ! [Xa: set_nat,A4: set_set_nat,X4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),A4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(X4),A4)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),X4)),A4) ) ) ).

% insert_subsetI
tff(fact_1771_insert__subsetI,axiom,
    ! [Xa: int,A4: set_int,X4: set_int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(X4),A4)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(insert_int(Xa),X4)),A4) ) ) ).

% insert_subsetI
tff(fact_1772_subset__Diff__insert,axiom,
    ! [A4: set_o,B4: set_o,Xa: $o,C3: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),minus_minus_set_o(B4,aa_set_o_set_o(insert_o((Xa)),C3)))
    <=> ( aa_set_o_o(ord_less_eq_set_o(A4),minus_minus_set_o(B4,C3))
        & ~ aa_set_o_o(member_o((Xa)),A4) ) ) ).

% subset_Diff_insert
tff(fact_1773_subset__Diff__insert,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,Xa: vEBT_VEBT,C3: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),minus_5127226145743854075T_VEBT(B4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),C3)))
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),minus_5127226145743854075T_VEBT(B4,C3))
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4) ) ) ).

% subset_Diff_insert
tff(fact_1774_subset__Diff__insert,axiom,
    ! [A4: set_real,B4: set_real,Xa: real,C3: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),minus_minus_set_real(B4,aa_set_real_set_real(insert_real(Xa),C3)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A4),minus_minus_set_real(B4,C3))
        & ~ aa_set_real_o(member_real(Xa),A4) ) ) ).

% subset_Diff_insert
tff(fact_1775_subset__Diff__insert,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,Xa: set_nat,C3: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),minus_2163939370556025621et_nat(B4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),C3)))
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),minus_2163939370556025621et_nat(B4,C3))
        & ~ aa_set_set_nat_o(member_set_nat(Xa),A4) ) ) ).

% subset_Diff_insert
tff(fact_1776_subset__Diff__insert,axiom,
    ! [A4: set_nat,B4: set_nat,Xa: nat,C3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),minus_minus_set_nat(B4,aa_set_nat_set_nat(insert_nat(Xa),C3)))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),minus_minus_set_nat(B4,C3))
        & ~ aa_set_nat_o(member_nat(Xa),A4) ) ) ).

% subset_Diff_insert
tff(fact_1777_subset__Diff__insert,axiom,
    ! [A4: set_int,B4: set_int,Xa: int,C3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),minus_minus_set_int(B4,aa_set_int_set_int(insert_int(Xa),C3)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A4),minus_minus_set_int(B4,C3))
        & ~ aa_set_int_o(member_int(Xa),A4) ) ) ).

% subset_Diff_insert
tff(fact_1778_finite_Ocases,axiom,
    ! [A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( ( A != bot_bo8194388402131092736T_VEBT )
       => ~ ! [A5: set_VEBT_VEBT] :
              ( ? [A3: vEBT_VEBT] : ( A = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A3),A5) )
             => ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A5) ) ) ) ).

% finite.cases
tff(fact_1779_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A5: set_complex] :
              ( ? [A3: complex] : ( A = insert_complex(A3,A5) )
             => ~ aa_set_complex_o(finite3207457112153483333omplex,A5) ) ) ) ).

% finite.cases
tff(fact_1780_finite_Ocases,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ~ ! [A5: set_Code_integer] :
              ( ? [A3: code_integer] : ( A = insert_Code_integer(A3,A5) )
             => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5) ) ) ) ).

% finite.cases
tff(fact_1781_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A5: set_nat] :
              ( ? [A3: nat] : ( A = aa_set_nat_set_nat(insert_nat(A3),A5) )
             => ~ aa_set_nat_o(finite_finite_nat,A5) ) ) ) ).

% finite.cases
tff(fact_1782_finite_Ocases,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ~ ! [A5: set_int] :
              ( ? [A3: int] : ( A = aa_set_int_set_int(insert_int(A3),A5) )
             => ~ aa_set_int_o(finite_finite_int,A5) ) ) ) ).

% finite.cases
tff(fact_1783_finite_Ocases,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ~ ! [A5: set_o] :
              ( ? [A3: $o] : ( A = aa_set_o_set_o(insert_o((A3)),A5) )
             => ~ aa_set_o_o(finite_finite_o,A5) ) ) ) ).

% finite.cases
tff(fact_1784_finite_Osimps,axiom,
    ! [A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
    <=> ( ( A = bot_bo8194388402131092736T_VEBT )
        | ? [A7: set_VEBT_VEBT,A6: vEBT_VEBT] :
            ( ( A = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A6),A7) )
            & aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A7) ) ) ) ).

% finite.simps
tff(fact_1785_finite_Osimps,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
    <=> ( ( A = bot_bot_set_complex )
        | ? [A7: set_complex,A6: complex] :
            ( ( A = insert_complex(A6,A7) )
            & aa_set_complex_o(finite3207457112153483333omplex,A7) ) ) ) ).

% finite.simps
tff(fact_1786_finite_Osimps,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
    <=> ( ( A = bot_bo3990330152332043303nteger )
        | ? [A7: set_Code_integer,A6: code_integer] :
            ( ( A = insert_Code_integer(A6,A7) )
            & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7) ) ) ) ).

% finite.simps
tff(fact_1787_finite_Osimps,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
    <=> ( ( A = bot_bot_set_nat )
        | ? [A7: set_nat,A6: nat] :
            ( ( A = aa_set_nat_set_nat(insert_nat(A6),A7) )
            & aa_set_nat_o(finite_finite_nat,A7) ) ) ) ).

% finite.simps
tff(fact_1788_finite_Osimps,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
    <=> ( ( A = bot_bot_set_int )
        | ? [A7: set_int,A6: int] :
            ( ( A = aa_set_int_set_int(insert_int(A6),A7) )
            & aa_set_int_o(finite_finite_int,A7) ) ) ) ).

% finite.simps
tff(fact_1789_finite_Osimps,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
    <=> ( ( A = bot_bot_set_o )
        | ? [A7: set_o,A6: $o] :
            ( ( A = aa_set_o_set_o(insert_o((A6)),A7) )
            & aa_set_o_o(finite_finite_o,A7) ) ) ) ).

% finite.simps
tff(fact_1790_finite__induct,axiom,
    ! [F2: set_VEBT_VEBT,P2: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F2)
     => ( aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT)
       => ( ! [X: vEBT_VEBT,F3: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F3)
             => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),F3)
               => ( aa_set_VEBT_VEBT_o(P2,F3)
                 => aa_set_VEBT_VEBT_o(P2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),F3)) ) ) )
         => aa_set_VEBT_VEBT_o(P2,F2) ) ) ) ).

% finite_induct
tff(fact_1791_finite__induct,axiom,
    ! [F2: set_real,P2: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F2)
     => ( aa_set_real_o(P2,bot_bot_set_real)
       => ( ! [X: real,F3: set_real] :
              ( aa_set_real_o(finite_finite_real,F3)
             => ( ~ aa_set_real_o(member_real(X),F3)
               => ( aa_set_real_o(P2,F3)
                 => aa_set_real_o(P2,aa_set_real_set_real(insert_real(X),F3)) ) ) )
         => aa_set_real_o(P2,F2) ) ) ) ).

% finite_induct
tff(fact_1792_finite__induct,axiom,
    ! [F2: set_set_nat,P2: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
     => ( aa_set_set_nat_o(P2,bot_bot_set_set_nat)
       => ( ! [X: set_nat,F3: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
             => ( ~ aa_set_set_nat_o(member_set_nat(X),F3)
               => ( aa_set_set_nat_o(P2,F3)
                 => aa_set_set_nat_o(P2,aa_set1013257548857649334et_nat(insert_set_nat(X),F3)) ) ) )
         => aa_set_set_nat_o(P2,F2) ) ) ) ).

% finite_induct
tff(fact_1793_finite__induct,axiom,
    ! [F2: set_complex,P2: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( aa_set_complex_o(P2,bot_bot_set_complex)
       => ( ! [X: complex,F3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F3)
             => ( ~ aa_set_complex_o(member_complex(X),F3)
               => ( aa_set_complex_o(P2,F3)
                 => aa_set_complex_o(P2,insert_complex(X,F3)) ) ) )
         => aa_set_complex_o(P2,F2) ) ) ) ).

% finite_induct
tff(fact_1794_finite__induct,axiom,
    ! [F2: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
       => ( ! [X: code_integer,F3: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
             => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),F3)
               => ( aa_set5643387332196575111eger_o(P2,F3)
                 => aa_set5643387332196575111eger_o(P2,insert_Code_integer(X,F3)) ) ) )
         => aa_set5643387332196575111eger_o(P2,F2) ) ) ) ).

% finite_induct
tff(fact_1795_finite__induct,axiom,
    ! [F2: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(P2,bot_bot_set_nat)
       => ( ! [X: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat(X),F3)
               => ( aa_set_nat_o(P2,F3)
                 => aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(X),F3)) ) ) )
         => aa_set_nat_o(P2,F2) ) ) ) ).

% finite_induct
tff(fact_1796_finite__induct,axiom,
    ! [F2: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(P2,bot_bot_set_int)
       => ( ! [X: int,F3: set_int] :
              ( aa_set_int_o(finite_finite_int,F3)
             => ( ~ aa_set_int_o(member_int(X),F3)
               => ( aa_set_int_o(P2,F3)
                 => aa_set_int_o(P2,aa_set_int_set_int(insert_int(X),F3)) ) ) )
         => aa_set_int_o(P2,F2) ) ) ) ).

% finite_induct
tff(fact_1797_finite__induct,axiom,
    ! [F2: set_o,P2: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(P2,bot_bot_set_o)
       => ( ! [X: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X)),F3)
               => ( aa_set_o_o(P2,F3)
                 => aa_set_o_o(P2,aa_set_o_set_o(insert_o((X)),F3)) ) ) )
         => aa_set_o_o(P2,F2) ) ) ) ).

% finite_induct
tff(fact_1798_finite__ne__induct,axiom,
    ! [F2: set_VEBT_VEBT,P2: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F2)
     => ( ( F2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X: vEBT_VEBT] : aa_set_VEBT_VEBT_o(P2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),bot_bo8194388402131092736T_VEBT))
         => ( ! [X: vEBT_VEBT,F3: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F3)
               => ( ( F3 != bot_bo8194388402131092736T_VEBT )
                 => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),F3)
                   => ( aa_set_VEBT_VEBT_o(P2,F3)
                     => aa_set_VEBT_VEBT_o(P2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),F3)) ) ) ) )
           => aa_set_VEBT_VEBT_o(P2,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1799_finite__ne__induct,axiom,
    ! [F2: set_real,P2: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F2)
     => ( ( F2 != bot_bot_set_real )
       => ( ! [X: real] : aa_set_real_o(P2,aa_set_real_set_real(insert_real(X),bot_bot_set_real))
         => ( ! [X: real,F3: set_real] :
                ( aa_set_real_o(finite_finite_real,F3)
               => ( ( F3 != bot_bot_set_real )
                 => ( ~ aa_set_real_o(member_real(X),F3)
                   => ( aa_set_real_o(P2,F3)
                     => aa_set_real_o(P2,aa_set_real_set_real(insert_real(X),F3)) ) ) ) )
           => aa_set_real_o(P2,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1800_finite__ne__induct,axiom,
    ! [F2: set_set_nat,P2: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
     => ( ( F2 != bot_bot_set_set_nat )
       => ( ! [X: set_nat] : aa_set_set_nat_o(P2,aa_set1013257548857649334et_nat(insert_set_nat(X),bot_bot_set_set_nat))
         => ( ! [X: set_nat,F3: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
               => ( ( F3 != bot_bot_set_set_nat )
                 => ( ~ aa_set_set_nat_o(member_set_nat(X),F3)
                   => ( aa_set_set_nat_o(P2,F3)
                     => aa_set_set_nat_o(P2,aa_set1013257548857649334et_nat(insert_set_nat(X),F3)) ) ) ) )
           => aa_set_set_nat_o(P2,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1801_finite__ne__induct,axiom,
    ! [F2: set_complex,P2: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( ( F2 != bot_bot_set_complex )
       => ( ! [X: complex] : aa_set_complex_o(P2,insert_complex(X,bot_bot_set_complex))
         => ( ! [X: complex,F3: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F3)
               => ( ( F3 != bot_bot_set_complex )
                 => ( ~ aa_set_complex_o(member_complex(X),F3)
                   => ( aa_set_complex_o(P2,F3)
                     => aa_set_complex_o(P2,insert_complex(X,F3)) ) ) ) )
           => aa_set_complex_o(P2,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1802_finite__ne__induct,axiom,
    ! [F2: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( ( F2 != bot_bo3990330152332043303nteger )
       => ( ! [X: code_integer] : aa_set5643387332196575111eger_o(P2,insert_Code_integer(X,bot_bo3990330152332043303nteger))
         => ( ! [X: code_integer,F3: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
               => ( ( F3 != bot_bo3990330152332043303nteger )
                 => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),F3)
                   => ( aa_set5643387332196575111eger_o(P2,F3)
                     => aa_set5643387332196575111eger_o(P2,insert_Code_integer(X,F3)) ) ) ) )
           => aa_set5643387332196575111eger_o(P2,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1803_finite__ne__induct,axiom,
    ! [F2: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X: nat] : aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))
         => ( ! [X: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(member_nat(X),F3)
                   => ( aa_set_nat_o(P2,F3)
                     => aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(X),F3)) ) ) ) )
           => aa_set_nat_o(P2,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1804_finite__ne__induct,axiom,
    ! [F2: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( ( F2 != bot_bot_set_int )
       => ( ! [X: int] : aa_set_int_o(P2,aa_set_int_set_int(insert_int(X),bot_bot_set_int))
         => ( ! [X: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( ( F3 != bot_bot_set_int )
                 => ( ~ aa_set_int_o(member_int(X),F3)
                   => ( aa_set_int_o(P2,F3)
                     => aa_set_int_o(P2,aa_set_int_set_int(insert_int(X),F3)) ) ) ) )
           => aa_set_int_o(P2,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1805_finite__ne__induct,axiom,
    ! [F2: set_o,P2: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( ( F2 != bot_bot_set_o )
       => ( ! [X: $o] : aa_set_o_o(P2,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))
         => ( ! [X: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( ( F3 != bot_bot_set_o )
                 => ( ~ aa_set_o_o(member_o((X)),F3)
                   => ( aa_set_o_o(P2,F3)
                     => aa_set_o_o(P2,aa_set_o_set_o(insert_o((X)),F3)) ) ) ) )
           => aa_set_o_o(P2,F2) ) ) ) ) ).

% finite_ne_induct
tff(fact_1806_infinite__finite__induct,axiom,
    ! [P2: fun_set_VEBT_VEBT_o,A4: set_VEBT_VEBT] :
      ( ! [A5: set_VEBT_VEBT] :
          ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A5)
         => aa_set_VEBT_VEBT_o(P2,A5) )
     => ( aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT)
       => ( ! [X: vEBT_VEBT,F3: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F3)
             => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),F3)
               => ( aa_set_VEBT_VEBT_o(P2,F3)
                 => aa_set_VEBT_VEBT_o(P2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),F3)) ) ) )
         => aa_set_VEBT_VEBT_o(P2,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_1807_infinite__finite__induct,axiom,
    ! [P2: fun_set_real_o,A4: set_real] :
      ( ! [A5: set_real] :
          ( ~ aa_set_real_o(finite_finite_real,A5)
         => aa_set_real_o(P2,A5) )
     => ( aa_set_real_o(P2,bot_bot_set_real)
       => ( ! [X: real,F3: set_real] :
              ( aa_set_real_o(finite_finite_real,F3)
             => ( ~ aa_set_real_o(member_real(X),F3)
               => ( aa_set_real_o(P2,F3)
                 => aa_set_real_o(P2,aa_set_real_set_real(insert_real(X),F3)) ) ) )
         => aa_set_real_o(P2,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_1808_infinite__finite__induct,axiom,
    ! [P2: fun_set_set_nat_o,A4: set_set_nat] :
      ( ! [A5: set_set_nat] :
          ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
         => aa_set_set_nat_o(P2,A5) )
     => ( aa_set_set_nat_o(P2,bot_bot_set_set_nat)
       => ( ! [X: set_nat,F3: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
             => ( ~ aa_set_set_nat_o(member_set_nat(X),F3)
               => ( aa_set_set_nat_o(P2,F3)
                 => aa_set_set_nat_o(P2,aa_set1013257548857649334et_nat(insert_set_nat(X),F3)) ) ) )
         => aa_set_set_nat_o(P2,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_1809_infinite__finite__induct,axiom,
    ! [P2: fun_set_complex_o,A4: set_complex] :
      ( ! [A5: set_complex] :
          ( ~ aa_set_complex_o(finite3207457112153483333omplex,A5)
         => aa_set_complex_o(P2,A5) )
     => ( aa_set_complex_o(P2,bot_bot_set_complex)
       => ( ! [X: complex,F3: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F3)
             => ( ~ aa_set_complex_o(member_complex(X),F3)
               => ( aa_set_complex_o(P2,F3)
                 => aa_set_complex_o(P2,insert_complex(X,F3)) ) ) )
         => aa_set_complex_o(P2,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_1810_infinite__finite__induct,axiom,
    ! [P2: fun_se7289222521248229150eger_o,A4: set_Code_integer] :
      ( ! [A5: set_Code_integer] :
          ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5)
         => aa_set5643387332196575111eger_o(P2,A5) )
     => ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
       => ( ! [X: code_integer,F3: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
             => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X),F3)
               => ( aa_set5643387332196575111eger_o(P2,F3)
                 => aa_set5643387332196575111eger_o(P2,insert_Code_integer(X,F3)) ) ) )
         => aa_set5643387332196575111eger_o(P2,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_1811_infinite__finite__induct,axiom,
    ! [P2: fun_set_nat_o,A4: set_nat] :
      ( ! [A5: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A5)
         => aa_set_nat_o(P2,A5) )
     => ( aa_set_nat_o(P2,bot_bot_set_nat)
       => ( ! [X: nat,F3: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F3)
             => ( ~ aa_set_nat_o(member_nat(X),F3)
               => ( aa_set_nat_o(P2,F3)
                 => aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(X),F3)) ) ) )
         => aa_set_nat_o(P2,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_1812_infinite__finite__induct,axiom,
    ! [P2: fun_set_int_o,A4: set_int] :
      ( ! [A5: set_int] :
          ( ~ aa_set_int_o(finite_finite_int,A5)
         => aa_set_int_o(P2,A5) )
     => ( aa_set_int_o(P2,bot_bot_set_int)
       => ( ! [X: int,F3: set_int] :
              ( aa_set_int_o(finite_finite_int,F3)
             => ( ~ aa_set_int_o(member_int(X),F3)
               => ( aa_set_int_o(P2,F3)
                 => aa_set_int_o(P2,aa_set_int_set_int(insert_int(X),F3)) ) ) )
         => aa_set_int_o(P2,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_1813_infinite__finite__induct,axiom,
    ! [P2: fun_set_o_o,A4: set_o] :
      ( ! [A5: set_o] :
          ( ~ aa_set_o_o(finite_finite_o,A5)
         => aa_set_o_o(P2,A5) )
     => ( aa_set_o_o(P2,bot_bot_set_o)
       => ( ! [X: $o,F3: set_o] :
              ( aa_set_o_o(finite_finite_o,F3)
             => ( ~ aa_set_o_o(member_o((X)),F3)
               => ( aa_set_o_o(P2,F3)
                 => aa_set_o_o(P2,aa_set_o_set_o(insert_o((X)),F3)) ) ) )
         => aa_set_o_o(P2,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_1814_infinite__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(S2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))) ) ).

% infinite_remove
tff(fact_1815_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ).

% infinite_remove
tff(fact_1816_infinite__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger))) ) ).

% infinite_remove
tff(fact_1817_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
     => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S2,aa_set_int_set_int(insert_int(A),bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_1818_infinite__remove,axiom,
    ! [S2: set_o,A: $o] :
      ( ~ aa_set_o_o(finite_finite_o,S2)
     => ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(S2,aa_set_o_set_o(insert_o((A)),bot_bot_set_o))) ) ).

% infinite_remove
tff(fact_1819_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S2,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_1820_infinite__coinduct,axiom,
    ! [X4: fun_set_VEBT_VEBT_o,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(X4,A4)
     => ( ! [A5: set_VEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(X4,A5)
           => ? [X3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),A5)
                & ( aa_set_VEBT_VEBT_o(X4,minus_5127226145743854075T_VEBT(A5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT)))
                  | ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) ) ) )
       => ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4) ) ) ).

% infinite_coinduct
tff(fact_1821_infinite__coinduct,axiom,
    ! [X4: fun_set_complex_o,A4: set_complex] :
      ( aa_set_complex_o(X4,A4)
     => ( ! [A5: set_complex] :
            ( aa_set_complex_o(X4,A5)
           => ? [X3: complex] :
                ( aa_set_complex_o(member_complex(X3),A5)
                & ( aa_set_complex_o(X4,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex)))
                  | ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex))) ) ) )
       => ~ aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% infinite_coinduct
tff(fact_1822_infinite__coinduct,axiom,
    ! [X4: fun_se7289222521248229150eger_o,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(X4,A4)
     => ( ! [A5: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(X4,A5)
           => ? [X3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A5)
                & ( aa_set5643387332196575111eger_o(X4,minus_2355218937544613996nteger(A5,insert_Code_integer(X3,bot_bo3990330152332043303nteger)))
                  | ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A5,insert_Code_integer(X3,bot_bo3990330152332043303nteger))) ) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ) ).

% infinite_coinduct
tff(fact_1823_infinite__coinduct,axiom,
    ! [X4: fun_set_int_o,A4: set_int] :
      ( aa_set_int_o(X4,A4)
     => ( ! [A5: set_int] :
            ( aa_set_int_o(X4,A5)
           => ? [X3: int] :
                ( aa_set_int_o(member_int(X3),A5)
                & ( aa_set_int_o(X4,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(X3),bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A4) ) ) ).

% infinite_coinduct
tff(fact_1824_infinite__coinduct,axiom,
    ! [X4: fun_set_o_o,A4: set_o] :
      ( aa_set_o_o(X4,A4)
     => ( ! [A5: set_o] :
            ( aa_set_o_o(X4,A5)
           => ? [X3: $o] :
                ( aa_set_o_o(member_o((X3)),A5)
                & ( aa_set_o_o(X4,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((X3)),bot_bot_set_o)))
                  | ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((X3)),bot_bot_set_o))) ) ) )
       => ~ aa_set_o_o(finite_finite_o,A4) ) ) ).

% infinite_coinduct
tff(fact_1825_infinite__coinduct,axiom,
    ! [X4: fun_set_nat_o,A4: set_nat] :
      ( aa_set_nat_o(X4,A4)
     => ( ! [A5: set_nat] :
            ( aa_set_nat_o(X4,A5)
           => ? [X3: nat] :
                ( aa_set_nat_o(member_nat(X3),A5)
                & ( aa_set_nat_o(X4,minus_minus_set_nat(A5,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A5,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% infinite_coinduct
tff(fact_1826_finite__empty__induct,axiom,
    ! [A4: set_VEBT_VEBT,P2: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_VEBT_VEBT_o(P2,A4)
       => ( ! [A3: vEBT_VEBT,A5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A5)
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A5)
               => ( aa_set_VEBT_VEBT_o(P2,A5)
                 => aa_set_VEBT_VEBT_o(P2,minus_5127226145743854075T_VEBT(A5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A3),bot_bo8194388402131092736T_VEBT))) ) ) )
         => aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% finite_empty_induct
tff(fact_1827_finite__empty__induct,axiom,
    ! [A4: set_real,P2: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(P2,A4)
       => ( ! [A3: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( aa_set_real_o(member_real(A3),A5)
               => ( aa_set_real_o(P2,A5)
                 => aa_set_real_o(P2,minus_minus_set_real(A5,aa_set_real_set_real(insert_real(A3),bot_bot_set_real))) ) ) )
         => aa_set_real_o(P2,bot_bot_set_real) ) ) ) ).

% finite_empty_induct
tff(fact_1828_finite__empty__induct,axiom,
    ! [A4: set_set_nat,P2: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(P2,A4)
       => ( ! [A3: set_nat,A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( aa_set_set_nat_o(member_set_nat(A3),A5)
               => ( aa_set_set_nat_o(P2,A5)
                 => aa_set_set_nat_o(P2,minus_2163939370556025621et_nat(A5,aa_set1013257548857649334et_nat(insert_set_nat(A3),bot_bot_set_set_nat))) ) ) )
         => aa_set_set_nat_o(P2,bot_bot_set_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_1829_finite__empty__induct,axiom,
    ! [A4: set_complex,P2: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(P2,A4)
       => ( ! [A3: complex,A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( aa_set_complex_o(member_complex(A3),A5)
               => ( aa_set_complex_o(P2,A5)
                 => aa_set_complex_o(P2,minus_811609699411566653omplex(A5,insert_complex(A3,bot_bot_set_complex))) ) ) )
         => aa_set_complex_o(P2,bot_bot_set_complex) ) ) ) ).

% finite_empty_induct
tff(fact_1830_finite__empty__induct,axiom,
    ! [A4: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(P2,A4)
       => ( ! [A3: code_integer,A5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5)
             => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A5)
               => ( aa_set5643387332196575111eger_o(P2,A5)
                 => aa_set5643387332196575111eger_o(P2,minus_2355218937544613996nteger(A5,insert_Code_integer(A3,bot_bo3990330152332043303nteger))) ) ) )
         => aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger) ) ) ) ).

% finite_empty_induct
tff(fact_1831_finite__empty__induct,axiom,
    ! [A4: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(P2,A4)
       => ( ! [A3: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( aa_set_int_o(member_int(A3),A5)
               => ( aa_set_int_o(P2,A5)
                 => aa_set_int_o(P2,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(A3),bot_bot_set_int))) ) ) )
         => aa_set_int_o(P2,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_1832_finite__empty__induct,axiom,
    ! [A4: set_o,P2: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(P2,A4)
       => ( ! [A3: $o,A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( aa_set_o_o(member_o((A3)),A5)
               => ( aa_set_o_o(P2,A5)
                 => aa_set_o_o(P2,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((A3)),bot_bot_set_o))) ) ) )
         => aa_set_o_o(P2,bot_bot_set_o) ) ) ) ).

% finite_empty_induct
tff(fact_1833_finite__empty__induct,axiom,
    ! [A4: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(P2,A4)
       => ( ! [A3: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( aa_set_nat_o(member_nat(A3),A5)
               => ( aa_set_nat_o(P2,A5)
                 => aa_set_nat_o(P2,minus_minus_set_nat(A5,aa_set_nat_set_nat(insert_nat(A3),bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P2,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_1834_subset__singleton__iff,axiom,
    ! [X4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(X4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))
    <=> ( ( X4 = bot_bo8194388402131092736T_VEBT )
        | ( X4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singleton_iff
tff(fact_1835_subset__singleton__iff,axiom,
    ! [X4: set_nat,A: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X4),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))
    <=> ( ( X4 = bot_bot_set_nat )
        | ( X4 = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_1836_subset__singleton__iff,axiom,
    ! [X4: set_o,A: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(X4),aa_set_o_set_o(insert_o((A)),bot_bot_set_o))
    <=> ( ( X4 = bot_bot_set_o )
        | ( X4 = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_1837_subset__singleton__iff,axiom,
    ! [X4: set_int,A: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X4),aa_set_int_set_int(insert_int(A),bot_bot_set_int))
    <=> ( ( X4 = bot_bot_set_int )
        | ( X4 = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_1838_subset__singletonD,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT))
     => ( ( A4 = bot_bo8194388402131092736T_VEBT )
        | ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singletonD
tff(fact_1839_subset__singletonD,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))
     => ( ( A4 = bot_bot_set_nat )
        | ( A4 = aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_1840_subset__singletonD,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))
     => ( ( A4 = bot_bot_set_o )
        | ( A4 = aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_1841_subset__singletonD,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))
     => ( ( A4 = bot_bot_set_int )
        | ( A4 = aa_set_int_set_int(insert_int(Xa),bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_1842_subset__insert__iff,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),B4))
    <=> $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT))),B4),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1843_subset__insert__iff,axiom,
    ! [A4: set_real,Xa: real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),aa_set_real_set_real(insert_real(Xa),B4))
    <=> $ite(aa_set_real_o(member_real(Xa),A4),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real))),B4),aa_set_real_o(ord_less_eq_set_real(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1844_subset__insert__iff,axiom,
    ! [A4: set_set_nat,Xa: set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),aa_set1013257548857649334et_nat(insert_set_nat(Xa),B4))
    <=> $ite(aa_set_set_nat_o(member_set_nat(Xa),A4),aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat))),B4),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1845_subset__insert__iff,axiom,
    ! [A4: set_o,Xa: $o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),aa_set_o_set_o(insert_o((Xa)),B4))
    <=> $ite(aa_set_o_o(member_o((Xa)),A4),aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))),B4),aa_set_o_o(ord_less_eq_set_o(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1846_subset__insert__iff,axiom,
    ! [A4: set_nat,Xa: nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(insert_nat(Xa),B4))
    <=> $ite(aa_set_nat_o(member_nat(Xa),A4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))),B4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)) ) ).

% subset_insert_iff
tff(fact_1847_subset__insert__iff,axiom,
    ! [A4: set_int,Xa: int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(insert_int(Xa),B4))
    <=> $ite(aa_set_int_o(member_int(Xa),A4),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))),B4),aa_set_int_o(ord_less_eq_set_int(A4),B4)) ) ).

% subset_insert_iff
tff(fact_1848_Diff__single__insert,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT))),B4)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),B4)) ) ).

% Diff_single_insert
tff(fact_1849_Diff__single__insert,axiom,
    ! [A4: set_o,Xa: $o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))),B4)
     => aa_set_o_o(ord_less_eq_set_o(A4),aa_set_o_set_o(insert_o((Xa)),B4)) ) ).

% Diff_single_insert
tff(fact_1850_Diff__single__insert,axiom,
    ! [A4: set_nat,Xa: nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(insert_nat(Xa),B4)) ) ).

% Diff_single_insert
tff(fact_1851_Diff__single__insert,axiom,
    ! [A4: set_int,Xa: int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))),B4)
     => aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(insert_int(Xa),B4)) ) ).

% Diff_single_insert
tff(fact_1852_remove__subset,axiom,
    ! [Xa: vEBT_VEBT,S2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),S2)
     => aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(S2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT))),S2) ) ).

% remove_subset
tff(fact_1853_remove__subset,axiom,
    ! [Xa: real,S2: set_real] :
      ( aa_set_real_o(member_real(Xa),S2)
     => aa_set_real_o(ord_less_set_real(minus_minus_set_real(S2,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real))),S2) ) ).

% remove_subset
tff(fact_1854_remove__subset,axiom,
    ! [Xa: set_nat,S2: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),S2)
     => aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(S2,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat))),S2) ) ).

% remove_subset
tff(fact_1855_remove__subset,axiom,
    ! [Xa: int,S2: set_int] :
      ( aa_set_int_o(member_int(Xa),S2)
     => aa_set_int_o(ord_less_set_int(minus_minus_set_int(S2,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))),S2) ) ).

% remove_subset
tff(fact_1856_remove__subset,axiom,
    ! [Xa: $o,S2: set_o] :
      ( aa_set_o_o(member_o((Xa)),S2)
     => aa_set_o_o(ord_less_set_o(minus_minus_set_o(S2,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))),S2) ) ).

% remove_subset
tff(fact_1857_remove__subset,axiom,
    ! [Xa: nat,S2: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),S2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,minus_minus_set_nat(S2,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))),S2) ) ).

% remove_subset
tff(fact_1858_finite__ranking__induct,axiom,
    ! [S2: set_VEBT_VEBT,P2: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT)
       => ( ! [X: vEBT_VEBT,S4: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
             => ( ! [Y4: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,Y4)),aa_VEBT_VEBT_rat(F,X)) )
               => ( aa_set_VEBT_VEBT_o(P2,S4)
                 => aa_set_VEBT_VEBT_o(P2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),S4)) ) ) )
         => aa_set_VEBT_VEBT_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1859_finite__ranking__induct,axiom,
    ! [S2: set_real,P2: fun_set_real_o,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(P2,bot_bot_set_real)
       => ( ! [X: real,S4: set_real] :
              ( aa_set_real_o(finite_finite_real,S4)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Y4)),aa_real_rat(F,X)) )
               => ( aa_set_real_o(P2,S4)
                 => aa_set_real_o(P2,aa_set_real_set_real(insert_real(X),S4)) ) ) )
         => aa_set_real_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1860_finite__ranking__induct,axiom,
    ! [S2: set_complex,P2: fun_set_complex_o,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P2,bot_bot_set_complex)
       => ( ! [X: complex,S4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S4)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,Y4)),aa_complex_rat(F,X)) )
               => ( aa_set_complex_o(P2,S4)
                 => aa_set_complex_o(P2,insert_complex(X,S4)) ) ) )
         => aa_set_complex_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1861_finite__ranking__induct,axiom,
    ! [S2: set_Code_integer,P2: fun_se7289222521248229150eger_o,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
       => ( ! [X: code_integer,S4: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S4)
             => ( ! [Y4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,Y4)),aa_Code_integer_rat(F,X)) )
               => ( aa_set5643387332196575111eger_o(P2,S4)
                 => aa_set5643387332196575111eger_o(P2,insert_Code_integer(X,S4)) ) ) )
         => aa_set5643387332196575111eger_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1862_finite__ranking__induct,axiom,
    ! [S2: set_nat,P2: fun_set_nat_o,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P2,bot_bot_set_nat)
       => ( ! [X: nat,S4: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S4)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Y4)),aa_nat_rat(F,X)) )
               => ( aa_set_nat_o(P2,S4)
                 => aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(X),S4)) ) ) )
         => aa_set_nat_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1863_finite__ranking__induct,axiom,
    ! [S2: set_int,P2: fun_set_int_o,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P2,bot_bot_set_int)
       => ( ! [X: int,S4: set_int] :
              ( aa_set_int_o(finite_finite_int,S4)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Y4)),aa_int_rat(F,X)) )
               => ( aa_set_int_o(P2,S4)
                 => aa_set_int_o(P2,aa_set_int_set_int(insert_int(X),S4)) ) ) )
         => aa_set_int_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1864_finite__ranking__induct,axiom,
    ! [S2: set_o,P2: fun_set_o_o,F: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_o_o(P2,bot_bot_set_o)
       => ( ! [X: $o,S4: set_o] :
              ( aa_set_o_o(finite_finite_o,S4)
             => ( ! [Y4: $o] :
                    ( aa_set_o_o(member_o((Y4)),S4)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(Y4))),aa_o_rat(F,(X))) )
               => ( aa_set_o_o(P2,S4)
                 => aa_set_o_o(P2,aa_set_o_set_o(insert_o((X)),S4)) ) ) )
         => aa_set_o_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1865_finite__ranking__induct,axiom,
    ! [S2: set_VEBT_VEBT,P2: fun_set_VEBT_VEBT_o,F: fun_VEBT_VEBT_num] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT)
       => ( ! [X: vEBT_VEBT,S4: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S4)
             => ( ! [Y4: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_VEBT_VEBT_num(F,Y4)),aa_VEBT_VEBT_num(F,X)) )
               => ( aa_set_VEBT_VEBT_o(P2,S4)
                 => aa_set_VEBT_VEBT_o(P2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),S4)) ) ) )
         => aa_set_VEBT_VEBT_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1866_finite__ranking__induct,axiom,
    ! [S2: set_real,P2: fun_set_real_o,F: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(P2,bot_bot_set_real)
       => ( ! [X: real,S4: set_real] :
              ( aa_set_real_o(finite_finite_real,S4)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,Y4)),aa_real_num(F,X)) )
               => ( aa_set_real_o(P2,S4)
                 => aa_set_real_o(P2,aa_set_real_set_real(insert_real(X),S4)) ) ) )
         => aa_set_real_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1867_finite__ranking__induct,axiom,
    ! [S2: set_complex,P2: fun_set_complex_o,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P2,bot_bot_set_complex)
       => ( ! [X: complex,S4: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S4)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S4)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,Y4)),aa_complex_num(F,X)) )
               => ( aa_set_complex_o(P2,S4)
                 => aa_set_complex_o(P2,insert_complex(X,S4)) ) ) )
         => aa_set_complex_o(P2,S2) ) ) ) ).

% finite_ranking_induct
tff(fact_1868_finite__linorder__max__induct,axiom,
    ! [A4: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
       => ( ! [B3: code_integer,A5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5)
             => ( ! [X3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A5)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),B3) )
               => ( aa_set5643387332196575111eger_o(P2,A5)
                 => aa_set5643387332196575111eger_o(P2,insert_Code_integer(B3,A5)) ) ) )
         => aa_set5643387332196575111eger_o(P2,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1869_finite__linorder__max__induct,axiom,
    ! [A4: set_o,P2: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(P2,bot_bot_set_o)
       => ( ! [B3: $o,A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( ! [X3: $o] :
                    ( aa_set_o_o(member_o((X3)),A5)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(X3)),(B3)) )
               => ( aa_set_o_o(P2,A5)
                 => aa_set_o_o(P2,aa_set_o_set_o(insert_o((B3)),A5)) ) ) )
         => aa_set_o_o(P2,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1870_finite__linorder__max__induct,axiom,
    ! [A4: set_real,P2: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(P2,bot_bot_set_real)
       => ( ! [B3: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),A5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B3) )
               => ( aa_set_real_o(P2,A5)
                 => aa_set_real_o(P2,aa_set_real_set_real(insert_real(B3),A5)) ) ) )
         => aa_set_real_o(P2,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1871_finite__linorder__max__induct,axiom,
    ! [A4: set_rat,P2: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(P2,bot_bot_set_rat)
       => ( ! [B3: rat,A5: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A5)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),B3) )
               => ( aa_set_rat_o(P2,A5)
                 => aa_set_rat_o(P2,insert_rat(B3,A5)) ) ) )
         => aa_set_rat_o(P2,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1872_finite__linorder__max__induct,axiom,
    ! [A4: set_num,P2: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(P2,bot_bot_set_num)
       => ( ! [B3: num,A5: set_num] :
              ( aa_set_num_o(finite_finite_num,A5)
             => ( ! [X3: num] :
                    ( aa_set_num_o(member_num(X3),A5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),B3) )
               => ( aa_set_num_o(P2,A5)
                 => aa_set_num_o(P2,insert_num(B3,A5)) ) ) )
         => aa_set_num_o(P2,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1873_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(P2,bot_bot_set_nat)
       => ( ! [B3: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),A5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),B3) )
               => ( aa_set_nat_o(P2,A5)
                 => aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(B3),A5)) ) ) )
         => aa_set_nat_o(P2,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1874_finite__linorder__max__induct,axiom,
    ! [A4: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(P2,bot_bot_set_int)
       => ( ! [B3: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),B3) )
               => ( aa_set_int_o(P2,A5)
                 => aa_set_int_o(P2,aa_set_int_set_int(insert_int(B3),A5)) ) ) )
         => aa_set_int_o(P2,A4) ) ) ) ).

% finite_linorder_max_induct
tff(fact_1875_finite__linorder__min__induct,axiom,
    ! [A4: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
       => ( ! [B3: code_integer,A5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5)
             => ( ! [X3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A5)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B3),X3) )
               => ( aa_set5643387332196575111eger_o(P2,A5)
                 => aa_set5643387332196575111eger_o(P2,insert_Code_integer(B3,A5)) ) ) )
         => aa_set5643387332196575111eger_o(P2,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1876_finite__linorder__min__induct,axiom,
    ! [A4: set_o,P2: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(P2,bot_bot_set_o)
       => ( ! [B3: $o,A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( ! [X3: $o] :
                    ( aa_set_o_o(member_o((X3)),A5)
                   => aa_o_o(aa_o_fun_o_o(ord_less_o,(B3)),(X3)) )
               => ( aa_set_o_o(P2,A5)
                 => aa_set_o_o(P2,aa_set_o_set_o(insert_o((B3)),A5)) ) ) )
         => aa_set_o_o(P2,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1877_finite__linorder__min__induct,axiom,
    ! [A4: set_real,P2: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(P2,bot_bot_set_real)
       => ( ! [B3: real,A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),A5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B3),X3) )
               => ( aa_set_real_o(P2,A5)
                 => aa_set_real_o(P2,aa_set_real_set_real(insert_real(B3),A5)) ) ) )
         => aa_set_real_o(P2,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1878_finite__linorder__min__induct,axiom,
    ! [A4: set_rat,P2: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(P2,bot_bot_set_rat)
       => ( ! [B3: rat,A5: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A5)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B3),X3) )
               => ( aa_set_rat_o(P2,A5)
                 => aa_set_rat_o(P2,insert_rat(B3,A5)) ) ) )
         => aa_set_rat_o(P2,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1879_finite__linorder__min__induct,axiom,
    ! [A4: set_num,P2: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(P2,bot_bot_set_num)
       => ( ! [B3: num,A5: set_num] :
              ( aa_set_num_o(finite_finite_num,A5)
             => ( ! [X3: num] :
                    ( aa_set_num_o(member_num(X3),A5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B3),X3) )
               => ( aa_set_num_o(P2,A5)
                 => aa_set_num_o(P2,insert_num(B3,A5)) ) ) )
         => aa_set_num_o(P2,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1880_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(P2,bot_bot_set_nat)
       => ( ! [B3: nat,A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),A5)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),X3) )
               => ( aa_set_nat_o(P2,A5)
                 => aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(B3),A5)) ) ) )
         => aa_set_nat_o(P2,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1881_finite__linorder__min__induct,axiom,
    ! [A4: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(P2,bot_bot_set_int)
       => ( ! [B3: int,A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A5)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B3),X3) )
               => ( aa_set_int_o(P2,A5)
                 => aa_set_int_o(P2,aa_set_int_set_int(insert_int(B3),A5)) ) ) )
         => aa_set_int_o(P2,A4) ) ) ) ).

% finite_linorder_min_induct
tff(fact_1882_finite__subset__induct,axiom,
    ! [F2: set_VEBT_VEBT,A4: set_VEBT_VEBT,P2: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F2),A4)
       => ( aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT)
         => ( ! [A3: vEBT_VEBT,F3: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F3)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A4)
                 => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),F3)
                   => ( aa_set_VEBT_VEBT_o(P2,F3)
                     => aa_set_VEBT_VEBT_o(P2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A3),F3)) ) ) ) )
           => aa_set_VEBT_VEBT_o(P2,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1883_finite__subset__induct,axiom,
    ! [F2: set_real,A4: set_real,P2: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F2)
     => ( aa_set_real_o(ord_less_eq_set_real(F2),A4)
       => ( aa_set_real_o(P2,bot_bot_set_real)
         => ( ! [A3: real,F3: set_real] :
                ( aa_set_real_o(finite_finite_real,F3)
               => ( aa_set_real_o(member_real(A3),A4)
                 => ( ~ aa_set_real_o(member_real(A3),F3)
                   => ( aa_set_real_o(P2,F3)
                     => aa_set_real_o(P2,aa_set_real_set_real(insert_real(A3),F3)) ) ) ) )
           => aa_set_real_o(P2,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1884_finite__subset__induct,axiom,
    ! [F2: set_set_nat,A4: set_set_nat,P2: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F2),A4)
       => ( aa_set_set_nat_o(P2,bot_bot_set_set_nat)
         => ( ! [A3: set_nat,F3: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
               => ( aa_set_set_nat_o(member_set_nat(A3),A4)
                 => ( ~ aa_set_set_nat_o(member_set_nat(A3),F3)
                   => ( aa_set_set_nat_o(P2,F3)
                     => aa_set_set_nat_o(P2,aa_set1013257548857649334et_nat(insert_set_nat(A3),F3)) ) ) ) )
           => aa_set_set_nat_o(P2,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1885_finite__subset__induct,axiom,
    ! [F2: set_complex,A4: set_complex,P2: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F2),A4)
       => ( aa_set_complex_o(P2,bot_bot_set_complex)
         => ( ! [A3: complex,F3: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F3)
               => ( aa_set_complex_o(member_complex(A3),A4)
                 => ( ~ aa_set_complex_o(member_complex(A3),F3)
                   => ( aa_set_complex_o(P2,F3)
                     => aa_set_complex_o(P2,insert_complex(A3,F3)) ) ) ) )
           => aa_set_complex_o(P2,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1886_finite__subset__induct,axiom,
    ! [F2: set_Code_integer,A4: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F2),A4)
       => ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F3: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
                 => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F3)
                   => ( aa_set5643387332196575111eger_o(P2,F3)
                     => aa_set5643387332196575111eger_o(P2,insert_Code_integer(A3,F3)) ) ) ) )
           => aa_set5643387332196575111eger_o(P2,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1887_finite__subset__induct,axiom,
    ! [F2: set_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F2),A4)
       => ( aa_set_nat_o(P2,bot_bot_set_nat)
         => ( ! [A3: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( aa_set_nat_o(member_nat(A3),A4)
                 => ( ~ aa_set_nat_o(member_nat(A3),F3)
                   => ( aa_set_nat_o(P2,F3)
                     => aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(A3),F3)) ) ) ) )
           => aa_set_nat_o(P2,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1888_finite__subset__induct,axiom,
    ! [F2: set_o,A4: set_o,P2: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(ord_less_eq_set_o(F2),A4)
       => ( aa_set_o_o(P2,bot_bot_set_o)
         => ( ! [A3: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( aa_set_o_o(member_o((A3)),A4)
                 => ( ~ aa_set_o_o(member_o((A3)),F3)
                   => ( aa_set_o_o(P2,F3)
                     => aa_set_o_o(P2,aa_set_o_set_o(insert_o((A3)),F3)) ) ) ) )
           => aa_set_o_o(P2,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1889_finite__subset__induct,axiom,
    ! [F2: set_int,A4: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(ord_less_eq_set_int(F2),A4)
       => ( aa_set_int_o(P2,bot_bot_set_int)
         => ( ! [A3: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( aa_set_int_o(member_int(A3),A4)
                 => ( ~ aa_set_int_o(member_int(A3),F3)
                   => ( aa_set_int_o(P2,F3)
                     => aa_set_int_o(P2,aa_set_int_set_int(insert_int(A3),F3)) ) ) ) )
           => aa_set_int_o(P2,F2) ) ) ) ) ).

% finite_subset_induct
tff(fact_1890_finite__subset__induct_H,axiom,
    ! [F2: set_VEBT_VEBT,A4: set_VEBT_VEBT,P2: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F2),A4)
       => ( aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT)
         => ( ! [A3: vEBT_VEBT,F3: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F3)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A4)
                 => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F3),A4)
                   => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),F3)
                     => ( aa_set_VEBT_VEBT_o(P2,F3)
                       => aa_set_VEBT_VEBT_o(P2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A3),F3)) ) ) ) ) )
           => aa_set_VEBT_VEBT_o(P2,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1891_finite__subset__induct_H,axiom,
    ! [F2: set_real,A4: set_real,P2: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F2)
     => ( aa_set_real_o(ord_less_eq_set_real(F2),A4)
       => ( aa_set_real_o(P2,bot_bot_set_real)
         => ( ! [A3: real,F3: set_real] :
                ( aa_set_real_o(finite_finite_real,F3)
               => ( aa_set_real_o(member_real(A3),A4)
                 => ( aa_set_real_o(ord_less_eq_set_real(F3),A4)
                   => ( ~ aa_set_real_o(member_real(A3),F3)
                     => ( aa_set_real_o(P2,F3)
                       => aa_set_real_o(P2,aa_set_real_set_real(insert_real(A3),F3)) ) ) ) ) )
           => aa_set_real_o(P2,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1892_finite__subset__induct_H,axiom,
    ! [F2: set_set_nat,A4: set_set_nat,P2: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F2),A4)
       => ( aa_set_set_nat_o(P2,bot_bot_set_set_nat)
         => ( ! [A3: set_nat,F3: set_set_nat] :
                ( aa_set_set_nat_o(finite1152437895449049373et_nat,F3)
               => ( aa_set_set_nat_o(member_set_nat(A3),A4)
                 => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(F3),A4)
                   => ( ~ aa_set_set_nat_o(member_set_nat(A3),F3)
                     => ( aa_set_set_nat_o(P2,F3)
                       => aa_set_set_nat_o(P2,aa_set1013257548857649334et_nat(insert_set_nat(A3),F3)) ) ) ) ) )
           => aa_set_set_nat_o(P2,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1893_finite__subset__induct_H,axiom,
    ! [F2: set_complex,A4: set_complex,P2: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F2),A4)
       => ( aa_set_complex_o(P2,bot_bot_set_complex)
         => ( ! [A3: complex,F3: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F3)
               => ( aa_set_complex_o(member_complex(A3),A4)
                 => ( aa_set_complex_o(ord_le211207098394363844omplex(F3),A4)
                   => ( ~ aa_set_complex_o(member_complex(A3),F3)
                     => ( aa_set_complex_o(P2,F3)
                       => aa_set_complex_o(P2,insert_complex(A3,F3)) ) ) ) ) )
           => aa_set_complex_o(P2,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1894_finite__subset__induct_H,axiom,
    ! [F2: set_Code_integer,A4: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F2),A4)
       => ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F3: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F3)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
                 => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F3),A4)
                   => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F3)
                     => ( aa_set5643387332196575111eger_o(P2,F3)
                       => aa_set5643387332196575111eger_o(P2,insert_Code_integer(A3,F3)) ) ) ) ) )
           => aa_set5643387332196575111eger_o(P2,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1895_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F2),A4)
       => ( aa_set_nat_o(P2,bot_bot_set_nat)
         => ( ! [A3: nat,F3: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F3)
               => ( aa_set_nat_o(member_nat(A3),A4)
                 => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,F3),A4)
                   => ( ~ aa_set_nat_o(member_nat(A3),F3)
                     => ( aa_set_nat_o(P2,F3)
                       => aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(A3),F3)) ) ) ) ) )
           => aa_set_nat_o(P2,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1896_finite__subset__induct_H,axiom,
    ! [F2: set_o,A4: set_o,P2: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F2)
     => ( aa_set_o_o(ord_less_eq_set_o(F2),A4)
       => ( aa_set_o_o(P2,bot_bot_set_o)
         => ( ! [A3: $o,F3: set_o] :
                ( aa_set_o_o(finite_finite_o,F3)
               => ( aa_set_o_o(member_o((A3)),A4)
                 => ( aa_set_o_o(ord_less_eq_set_o(F3),A4)
                   => ( ~ aa_set_o_o(member_o((A3)),F3)
                     => ( aa_set_o_o(P2,F3)
                       => aa_set_o_o(P2,aa_set_o_set_o(insert_o((A3)),F3)) ) ) ) ) )
           => aa_set_o_o(P2,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1897_finite__subset__induct_H,axiom,
    ! [F2: set_int,A4: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( aa_set_int_o(ord_less_eq_set_int(F2),A4)
       => ( aa_set_int_o(P2,bot_bot_set_int)
         => ( ! [A3: int,F3: set_int] :
                ( aa_set_int_o(finite_finite_int,F3)
               => ( aa_set_int_o(member_int(A3),A4)
                 => ( aa_set_int_o(ord_less_eq_set_int(F3),A4)
                   => ( ~ aa_set_int_o(member_int(A3),F3)
                     => ( aa_set_int_o(P2,F3)
                       => aa_set_int_o(P2,aa_set_int_set_int(insert_int(A3),F3)) ) ) ) ) )
           => aa_set_int_o(P2,F2) ) ) ) ) ).

% finite_subset_induct'
tff(fact_1898_remove__induct,axiom,
    ! [P2: fun_set_VEBT_VEBT_o,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT)
     => ( ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
         => aa_set_VEBT_VEBT_o(P2,B4) )
       => ( ! [A5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A5)
             => ( ( A5 != bot_bo8194388402131092736T_VEBT )
               => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A5),B4)
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),A5)
                       => aa_set_VEBT_VEBT_o(P2,minus_5127226145743854075T_VEBT(A5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) )
                   => aa_set_VEBT_VEBT_o(P2,A5) ) ) ) )
         => aa_set_VEBT_VEBT_o(P2,B4) ) ) ) ).

% remove_induct
tff(fact_1899_remove__induct,axiom,
    ! [P2: fun_set_real_o,B4: set_real] :
      ( aa_set_real_o(P2,bot_bot_set_real)
     => ( ( ~ aa_set_real_o(finite_finite_real,B4)
         => aa_set_real_o(P2,B4) )
       => ( ! [A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ( A5 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A5),B4)
                 => ( ! [X3: real] :
                        ( aa_set_real_o(member_real(X3),A5)
                       => aa_set_real_o(P2,minus_minus_set_real(A5,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) )
                   => aa_set_real_o(P2,A5) ) ) ) )
         => aa_set_real_o(P2,B4) ) ) ) ).

% remove_induct
tff(fact_1900_remove__induct,axiom,
    ! [P2: fun_set_set_nat_o,B4: set_set_nat] :
      ( aa_set_set_nat_o(P2,bot_bot_set_set_nat)
     => ( ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
         => aa_set_set_nat_o(P2,B4) )
       => ( ! [A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( ( A5 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A5),B4)
                 => ( ! [X3: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X3),A5)
                       => aa_set_set_nat_o(P2,minus_2163939370556025621et_nat(A5,aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P2,A5) ) ) ) )
         => aa_set_set_nat_o(P2,B4) ) ) ) ).

% remove_induct
tff(fact_1901_remove__induct,axiom,
    ! [P2: fun_set_complex_o,B4: set_complex] :
      ( aa_set_complex_o(P2,bot_bot_set_complex)
     => ( ( ~ aa_set_complex_o(finite3207457112153483333omplex,B4)
         => aa_set_complex_o(P2,B4) )
       => ( ! [A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( ( A5 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A5),B4)
                 => ( ! [X3: complex] :
                        ( aa_set_complex_o(member_complex(X3),A5)
                       => aa_set_complex_o(P2,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex))) )
                   => aa_set_complex_o(P2,A5) ) ) ) )
         => aa_set_complex_o(P2,B4) ) ) ) ).

% remove_induct
tff(fact_1902_remove__induct,axiom,
    ! [P2: fun_se7289222521248229150eger_o,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
     => ( ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => aa_set5643387332196575111eger_o(P2,B4) )
       => ( ! [A5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5)
             => ( ( A5 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A5),B4)
                 => ( ! [X3: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A5)
                       => aa_set5643387332196575111eger_o(P2,minus_2355218937544613996nteger(A5,insert_Code_integer(X3,bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(P2,A5) ) ) ) )
         => aa_set5643387332196575111eger_o(P2,B4) ) ) ) ).

% remove_induct
tff(fact_1903_remove__induct,axiom,
    ! [P2: fun_set_o_o,B4: set_o] :
      ( aa_set_o_o(P2,bot_bot_set_o)
     => ( ( ~ aa_set_o_o(finite_finite_o,B4)
         => aa_set_o_o(P2,B4) )
       => ( ! [A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( ( A5 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A5),B4)
                 => ( ! [X3: $o] :
                        ( aa_set_o_o(member_o((X3)),A5)
                       => aa_set_o_o(P2,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((X3)),bot_bot_set_o))) )
                   => aa_set_o_o(P2,A5) ) ) ) )
         => aa_set_o_o(P2,B4) ) ) ) ).

% remove_induct
tff(fact_1904_remove__induct,axiom,
    ! [P2: fun_set_nat_o,B4: set_nat] :
      ( aa_set_nat_o(P2,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B4)
         => aa_set_nat_o(P2,B4) )
       => ( ! [A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ( A5 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),B4)
                 => ( ! [X3: nat] :
                        ( aa_set_nat_o(member_nat(X3),A5)
                       => aa_set_nat_o(P2,minus_minus_set_nat(A5,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) )
                   => aa_set_nat_o(P2,A5) ) ) ) )
         => aa_set_nat_o(P2,B4) ) ) ) ).

% remove_induct
tff(fact_1905_remove__induct,axiom,
    ! [P2: fun_set_int_o,B4: set_int] :
      ( aa_set_int_o(P2,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B4)
         => aa_set_int_o(P2,B4) )
       => ( ! [A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ( A5 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A5),B4)
                 => ( ! [X3: int] :
                        ( aa_set_int_o(member_int(X3),A5)
                       => aa_set_int_o(P2,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) )
                   => aa_set_int_o(P2,A5) ) ) ) )
         => aa_set_int_o(P2,B4) ) ) ) ).

% remove_induct
tff(fact_1906_finite__remove__induct,axiom,
    ! [B4: set_VEBT_VEBT,P2: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT)
       => ( ! [A5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A5)
             => ( ( A5 != bot_bo8194388402131092736T_VEBT )
               => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A5),B4)
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),A5)
                       => aa_set_VEBT_VEBT_o(P2,minus_5127226145743854075T_VEBT(A5,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),bot_bo8194388402131092736T_VEBT))) )
                   => aa_set_VEBT_VEBT_o(P2,A5) ) ) ) )
         => aa_set_VEBT_VEBT_o(P2,B4) ) ) ) ).

% finite_remove_induct
tff(fact_1907_finite__remove__induct,axiom,
    ! [B4: set_real,P2: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(P2,bot_bot_set_real)
       => ( ! [A5: set_real] :
              ( aa_set_real_o(finite_finite_real,A5)
             => ( ( A5 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A5),B4)
                 => ( ! [X3: real] :
                        ( aa_set_real_o(member_real(X3),A5)
                       => aa_set_real_o(P2,minus_minus_set_real(A5,aa_set_real_set_real(insert_real(X3),bot_bot_set_real))) )
                   => aa_set_real_o(P2,A5) ) ) ) )
         => aa_set_real_o(P2,B4) ) ) ) ).

% finite_remove_induct
tff(fact_1908_finite__remove__induct,axiom,
    ! [B4: set_set_nat,P2: fun_set_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(P2,bot_bot_set_set_nat)
       => ( ! [A5: set_set_nat] :
              ( aa_set_set_nat_o(finite1152437895449049373et_nat,A5)
             => ( ( A5 != bot_bot_set_set_nat )
               => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A5),B4)
                 => ( ! [X3: set_nat] :
                        ( aa_set_set_nat_o(member_set_nat(X3),A5)
                       => aa_set_set_nat_o(P2,minus_2163939370556025621et_nat(A5,aa_set1013257548857649334et_nat(insert_set_nat(X3),bot_bot_set_set_nat))) )
                   => aa_set_set_nat_o(P2,A5) ) ) ) )
         => aa_set_set_nat_o(P2,B4) ) ) ) ).

% finite_remove_induct
tff(fact_1909_finite__remove__induct,axiom,
    ! [B4: set_complex,P2: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(P2,bot_bot_set_complex)
       => ( ! [A5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A5)
             => ( ( A5 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A5),B4)
                 => ( ! [X3: complex] :
                        ( aa_set_complex_o(member_complex(X3),A5)
                       => aa_set_complex_o(P2,minus_811609699411566653omplex(A5,insert_complex(X3,bot_bot_set_complex))) )
                   => aa_set_complex_o(P2,A5) ) ) ) )
         => aa_set_complex_o(P2,B4) ) ) ) ).

% finite_remove_induct
tff(fact_1910_finite__remove__induct,axiom,
    ! [B4: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
       => ( ! [A5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A5)
             => ( ( A5 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A5),B4)
                 => ( ! [X3: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A5)
                       => aa_set5643387332196575111eger_o(P2,minus_2355218937544613996nteger(A5,insert_Code_integer(X3,bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(P2,A5) ) ) ) )
         => aa_set5643387332196575111eger_o(P2,B4) ) ) ) ).

% finite_remove_induct
tff(fact_1911_finite__remove__induct,axiom,
    ! [B4: set_o,P2: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,B4)
     => ( aa_set_o_o(P2,bot_bot_set_o)
       => ( ! [A5: set_o] :
              ( aa_set_o_o(finite_finite_o,A5)
             => ( ( A5 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A5),B4)
                 => ( ! [X3: $o] :
                        ( aa_set_o_o(member_o((X3)),A5)
                       => aa_set_o_o(P2,minus_minus_set_o(A5,aa_set_o_set_o(insert_o((X3)),bot_bot_set_o))) )
                   => aa_set_o_o(P2,A5) ) ) ) )
         => aa_set_o_o(P2,B4) ) ) ) ).

% finite_remove_induct
tff(fact_1912_finite__remove__induct,axiom,
    ! [B4: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(P2,bot_bot_set_nat)
       => ( ! [A5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A5)
             => ( ( A5 != bot_bot_set_nat )
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A5),B4)
                 => ( ! [X3: nat] :
                        ( aa_set_nat_o(member_nat(X3),A5)
                       => aa_set_nat_o(P2,minus_minus_set_nat(A5,aa_set_nat_set_nat(insert_nat(X3),bot_bot_set_nat))) )
                   => aa_set_nat_o(P2,A5) ) ) ) )
         => aa_set_nat_o(P2,B4) ) ) ) ).

% finite_remove_induct
tff(fact_1913_finite__remove__induct,axiom,
    ! [B4: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(P2,bot_bot_set_int)
       => ( ! [A5: set_int] :
              ( aa_set_int_o(finite_finite_int,A5)
             => ( ( A5 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A5),B4)
                 => ( ! [X3: int] :
                        ( aa_set_int_o(member_int(X3),A5)
                       => aa_set_int_o(P2,minus_minus_set_int(A5,aa_set_int_set_int(insert_int(X3),bot_bot_set_int))) )
                   => aa_set_int_o(P2,A5) ) ) ) )
         => aa_set_int_o(P2,B4) ) ) ) ).

% finite_remove_induct
tff(fact_1914_finite__induct__select,axiom,
    ! [S2: set_VEBT_VEBT,P2: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_VEBT_VEBT_o(P2,bot_bo8194388402131092736T_VEBT)
       => ( ! [T4: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(T4),S2)
             => ( aa_set_VEBT_VEBT_o(P2,T4)
               => ? [X3: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),minus_5127226145743854075T_VEBT(S2,T4))
                    & aa_set_VEBT_VEBT_o(P2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X3),T4)) ) ) )
         => aa_set_VEBT_VEBT_o(P2,S2) ) ) ) ).

% finite_induct_select
tff(fact_1915_finite__induct__select,axiom,
    ! [S2: set_complex,P2: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_complex_o(P2,bot_bot_set_complex)
       => ( ! [T4: set_complex] :
              ( aa_set_complex_o(ord_less_set_complex(T4),S2)
             => ( aa_set_complex_o(P2,T4)
               => ? [X3: complex] :
                    ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(S2,T4))
                    & aa_set_complex_o(P2,insert_complex(X3,T4)) ) ) )
         => aa_set_complex_o(P2,S2) ) ) ) ).

% finite_induct_select
tff(fact_1916_finite__induct__select,axiom,
    ! [S2: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( aa_set5643387332196575111eger_o(P2,bot_bo3990330152332043303nteger)
       => ( ! [T4: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(T4),S2)
             => ( aa_set5643387332196575111eger_o(P2,T4)
               => ? [X3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(S2,T4))
                    & aa_set5643387332196575111eger_o(P2,insert_Code_integer(X3,T4)) ) ) )
         => aa_set5643387332196575111eger_o(P2,S2) ) ) ) ).

% finite_induct_select
tff(fact_1917_finite__induct__select,axiom,
    ! [S2: set_int,P2: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_int_o(P2,bot_bot_set_int)
       => ( ! [T4: set_int] :
              ( aa_set_int_o(ord_less_set_int(T4),S2)
             => ( aa_set_int_o(P2,T4)
               => ? [X3: int] :
                    ( aa_set_int_o(member_int(X3),minus_minus_set_int(S2,T4))
                    & aa_set_int_o(P2,aa_set_int_set_int(insert_int(X3),T4)) ) ) )
         => aa_set_int_o(P2,S2) ) ) ) ).

% finite_induct_select
tff(fact_1918_finite__induct__select,axiom,
    ! [S2: set_o,P2: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( aa_set_o_o(P2,bot_bot_set_o)
       => ( ! [T4: set_o] :
              ( aa_set_o_o(ord_less_set_o(T4),S2)
             => ( aa_set_o_o(P2,T4)
               => ? [X3: $o] :
                    ( aa_set_o_o(member_o((X3)),minus_minus_set_o(S2,T4))
                    & aa_set_o_o(P2,aa_set_o_set_o(insert_o((X3)),T4)) ) ) )
         => aa_set_o_o(P2,S2) ) ) ) ).

% finite_induct_select
tff(fact_1919_finite__induct__select,axiom,
    ! [S2: set_nat,P2: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(P2,bot_bot_set_nat)
       => ( ! [T4: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,T4),S2)
             => ( aa_set_nat_o(P2,T4)
               => ? [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(S2,T4))
                    & aa_set_nat_o(P2,aa_set_nat_set_nat(insert_nat(X3),T4)) ) ) )
         => aa_set_nat_o(P2,S2) ) ) ) ).

% finite_induct_select
tff(fact_1920_psubset__insert__iff,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),B4))
    <=> $ite(
          aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),B4),
          aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A4),B4),
          $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4),aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT))),B4),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1921_psubset__insert__iff,axiom,
    ! [A4: set_real,Xa: real,B4: set_real] :
      ( aa_set_real_o(ord_less_set_real(A4),aa_set_real_set_real(insert_real(Xa),B4))
    <=> $ite(
          aa_set_real_o(member_real(Xa),B4),
          aa_set_real_o(ord_less_set_real(A4),B4),
          $ite(aa_set_real_o(member_real(Xa),A4),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real))),B4),aa_set_real_o(ord_less_eq_set_real(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1922_psubset__insert__iff,axiom,
    ! [A4: set_set_nat,Xa: set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A4),aa_set1013257548857649334et_nat(insert_set_nat(Xa),B4))
    <=> $ite(
          aa_set_set_nat_o(member_set_nat(Xa),B4),
          aa_set_set_nat_o(ord_less_set_set_nat(A4),B4),
          $ite(aa_set_set_nat_o(member_set_nat(Xa),A4),aa_set_set_nat_o(ord_less_set_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat))),B4),aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1923_psubset__insert__iff,axiom,
    ! [A4: set_o,Xa: $o,B4: set_o] :
      ( aa_set_o_o(ord_less_set_o(A4),aa_set_o_set_o(insert_o((Xa)),B4))
    <=> $ite(
          aa_set_o_o(member_o((Xa)),B4),
          aa_set_o_o(ord_less_set_o(A4),B4),
          $ite(aa_set_o_o(member_o((Xa)),A4),aa_set_o_o(ord_less_set_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))),B4),aa_set_o_o(ord_less_eq_set_o(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1924_psubset__insert__iff,axiom,
    ! [A4: set_nat,Xa: nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),aa_set_nat_set_nat(insert_nat(Xa),B4))
    <=> $ite(
          aa_set_nat_o(member_nat(Xa),B4),
          aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4),
          $ite(aa_set_nat_o(member_nat(Xa),A4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))),B4),aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1925_psubset__insert__iff,axiom,
    ! [A4: set_int,Xa: int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),aa_set_int_set_int(insert_int(Xa),B4))
    <=> $ite(
          aa_set_int_o(member_int(Xa),B4),
          aa_set_int_o(ord_less_set_int(A4),B4),
          $ite(aa_set_int_o(member_int(Xa),A4),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))),B4),aa_set_int_o(ord_less_eq_set_int(A4),B4)) ) ) ).

% psubset_insert_iff
tff(fact_1926_finite__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ? [Xs2: list_VEBT_VEBT] : ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs2) = A4 ) ) ).

% finite_list
tff(fact_1927_finite__list,axiom,
    ! [A4: set_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ? [Xs2: list_real] : ( aa_lis3769085826154824284t_real(set_real2,Xs2) = A4 ) ) ).

% finite_list
tff(fact_1928_finite__list,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ? [Xs2: list_o] : ( aa_list_o_set_o(set_o2,Xs2) = A4 ) ) ).

% finite_list
tff(fact_1929_finite__list,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ? [Xs2: list_nat] : ( aa_list_nat_set_nat(set_nat2,Xs2) = A4 ) ) ).

% finite_list
tff(fact_1930_finite__list,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ? [Xs2: list_int] : ( aa_list_int_set_int(set_int2,Xs2) = A4 ) ) ).

% finite_list
tff(fact_1931_finite__list,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ? [Xs2: list_complex] : ( set_complex2(Xs2) = A4 ) ) ).

% finite_list
tff(fact_1932_finite__list,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ? [Xs2: list_Code_integer] : ( set_Code_integer2(Xs2) = A4 ) ) ).

% finite_list
tff(fact_1933_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),B4)
    <=> ! [X5: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X5),set_set_nat2(Xs))
         => aa_set_set_nat_o(member_set_nat(X5),B4) ) ) ).

% subset_code(1)
tff(fact_1934_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),B4)
    <=> ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),B4) ) ) ).

% subset_code(1)
tff(fact_1935_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Xs)),B4)
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_list_nat_set_nat(set_nat2,Xs))
         => aa_set_nat_o(member_nat(X5),B4) ) ) ).

% subset_code(1)
tff(fact_1936_subset__code_I1_J,axiom,
    ! [Xs: list_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),B4)
    <=> ! [X5: real] :
          ( aa_set_real_o(member_real(X5),aa_lis3769085826154824284t_real(set_real2,Xs))
         => aa_set_real_o(member_real(X5),B4) ) ) ).

% subset_code(1)
tff(fact_1937_subset__code_I1_J,axiom,
    ! [Xs: list_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Xs)),B4)
    <=> ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),aa_list_o_set_o(set_o2,Xs))
         => aa_set_o_o(member_o((X5)),B4) ) ) ).

% subset_code(1)
tff(fact_1938_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Xs)),B4)
    <=> ! [X5: int] :
          ( aa_set_int_o(member_int(X5),aa_list_int_set_int(set_int2,Xs))
         => aa_set_int_o(member_int(X5),B4) ) ) ).

% subset_code(1)
tff(fact_1939_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Xa)
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),Xa) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_1940_pred__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Px: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some_nat(Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),Xa),Px) ) ) ).

% pred_corr
tff(fact_1941_pred__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(T)),Xa),Sx) ) ) ).

% pred_correct
tff(fact_1942_set__vebt__pred,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Px: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some_nat(Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(T)),Xa),Px) ) ) ).

% set_vebt_pred
tff(fact_1943_succ__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_succ(T,Xa) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),Xa),Sx) ) ) ).

% succ_corr
tff(fact_1944_succ__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_succ(T,Xa) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(T)),Xa),Sx) ) ) ).

% succ_correct
tff(fact_1945_maxt__sound,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),Xa)
       => ( vEBT_vebt_maxt(T) = some_nat(Xa) ) ) ) ).

% maxt_sound
tff(fact_1946_maxt__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Xa) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),Xa) ) ) ).

% maxt_corr
tff(fact_1947_set__vebt__maxt,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Xa) )
      <=> aa_nat_o(vEBT_VEBT_max_in_set(vEBT_set_vebt(T)),Xa) ) ) ).

% set_vebt_maxt
tff(fact_1948_mint__sound,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),Xa)
       => ( vEBT_vebt_mint(T) = some_nat(Xa) ) ) ) ).

% mint_sound
tff(fact_1949_maxbmo,axiom,
    ! [T: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_maxt(T) = some_nat(Xa) )
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xa) ) ).

% maxbmo
tff(fact_1950_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => ( vEBT_vebt_mint(T) = none_nat ) ) ).

% minNullmin
tff(fact_1951_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T) ) ).

% minminNull
tff(fact_1952_vebt__minNull__mint,axiom,
    ! [T: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
    <=> ( vEBT_vebt_mint(T) = none_nat ) ) ).

% vebt_minNull_mint
tff(fact_1953_mint__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Maxi) ) ) ).

% mint_member
tff(fact_1954_maxt__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Maxi) ) ) ).

% maxt_member
tff(fact_1955_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,Na: nat,Mini: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xa)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xa) ) ) ) ).

% mint_corr_help
tff(fact_1956_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,Na: nat,Maxi: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Xa)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_1957_set__vebt__mint_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Xa) )
      <=> ( aa_set_nat_o(member_nat(Xa),vEBT_set_vebt(T))
          & ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),vEBT_set_vebt(T))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X5) ) ) ) ) ).

% set_vebt_mint'
tff(fact_1958_set__vebt__maxt_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_maxt(T) = some_nat(Xa) )
      <=> ( aa_set_nat_o(member_nat(Xa),vEBT_set_vebt(T))
          & ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),vEBT_set_vebt(T))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa) ) ) ) ) ).

% set_vebt_maxt'
tff(fact_1959_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_mint(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_1960_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_maxt(T) = none_nat )
       => ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_1961_set__vebt__mint,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Xa) )
      <=> aa_nat_o(vEBT_VEBT_min_in_set(vEBT_set_vebt(T)),Xa) ) ) ).

% set_vebt_mint
tff(fact_1962_mint__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_mint(T) = some_nat(Xa) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),Xa) ) ) ).

% mint_corr
tff(fact_1963_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Na: nat] : ( vEBT_vebt_succ(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,Na)) = none_nat ) ).

% vebt_succ.simps(2)
tff(fact_1964_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : ( vEBT_vebt_pred(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat) = none_nat ) ).

% vebt_pred.simps(1)
tff(fact_1965_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz),Va2) = none_nat ) ).

% vebt_succ.simps(3)
tff(fact_1966_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz),Va2),Vb) = none_nat ) ).

% vebt_pred.simps(4)
tff(fact_1967_vebt__succ_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2),Ve2) = none_nat ) ).

% vebt_succ.simps(4)
tff(fact_1968_vebt__pred_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2),Vf2) = none_nat ) ).

% vebt_pred.simps(5)
tff(fact_1969_vebt__succ_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Vi2) = none_nat ) ).

% vebt_succ.simps(5)
tff(fact_1970_vebt__pred_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Vj2) = none_nat ) ).

% vebt_pred.simps(6)
tff(fact_1971_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] :
      ( vEBT_vebt_succ(vEBT_Leaf((Uu2),(B)),zero_zero_nat) = $ite((B),some_nat(one_one_nat),none_nat) ) ).

% vebt_succ.simps(1)
tff(fact_1972_vebt__succ__code_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( vEBT_vebt_succ(vEBT_Leaf((A),(B)),Xa) = $ite(
          ( (B)
          & ( Xa = zero_zero_nat ) ),
          some_nat(one_one_nat),
          none_nat ) ) ).

% vebt_succ_code(1)
tff(fact_1973_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( vEBT_vebt_pred(vEBT_Leaf((A),(Uw2)),aa_nat_nat(suc,zero_zero_nat)) = $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(2)
tff(fact_1974_vebt__pred__code,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( vEBT_vebt_pred(vEBT_Leaf((A),(B)),Xa) = $ite(
          Xa = zero_zero_nat,
          none_nat,
          $ite(
            Xa = one_one_nat,
            $ite((A),some_nat(zero_zero_nat),none_nat),
            $ite(
              (B),
              some_nat(one_one_nat),
              $ite((A),some_nat(zero_zero_nat),none_nat) ) ) ) ) ).

% vebt_pred_code
tff(fact_1975_vebt__pred_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va2: nat] :
      ( vEBT_vebt_pred(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
          (B),
          some_nat(one_one_nat),
          $ite((A),some_nat(zero_zero_nat),none_nat) ) ) ).

% vebt_pred.simps(3)
tff(fact_1976_vebt__maxt_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  (B3),
                  some_nat(one_one_nat),
                  $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2) )
               => ( Y != some_nat(Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_1977_vebt__mint_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  (A3),
                  some_nat(zero_zero_nat),
                  $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2) )
               => ( Y != some_nat(Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_1978_vebt__maxt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( vEBT_vebt_maxt(vEBT_Leaf((A),(B))) = $ite(
          (B),
          some_nat(one_one_nat),
          $ite((A),some_nat(zero_zero_nat),none_nat) ) ) ).

% vebt_maxt.simps(1)
tff(fact_1979_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( vEBT_vebt_mint(vEBT_Leaf((A),(B))) = $ite(
          (A),
          some_nat(zero_zero_nat),
          $ite((B),some_nat(one_one_nat),none_nat) ) ) ).

% vebt_mint.simps(1)
tff(fact_1980_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : ( vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz)) = some_nat(Ma) ) ).

% vebt_maxt.simps(3)
tff(fact_1981_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : ( vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz)) = some_nat(Mi) ) ).

% vebt_mint.simps(3)
tff(fact_1982_set__vebt__mint_H_H,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( vEBT_vebt_mint(T) = $ite(vEBT_set_vebt(T) = bot_bot_set_nat,none_nat,some_nat(aa_set_nat_nat(lattic8721135487736765967in_nat,vEBT_set_vebt(T)))) ) ) ).

% set_vebt_mint''
tff(fact_1983_set__vebt__maxt_H_H,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( vEBT_vebt_maxt(T) = $ite(vEBT_set_vebt(T) = bot_bot_set_nat,none_nat,some_nat(aa_set_nat_nat(lattic8265883725875713057ax_nat,vEBT_set_vebt(T)))) ) ) ).

% set_vebt_maxt''
tff(fact_1984_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : ( vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) = none_nat ) ).

% vebt_maxt.simps(2)
tff(fact_1985_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : ( vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) = none_nat ) ).

% vebt_mint.simps(2)
tff(fact_1986_Max__singleton,axiom,
    ! [Xa: int] : ( lattic8263393255366662781ax_int(aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) = Xa ) ).

% Max_singleton
tff(fact_1987_Max__singleton,axiom,
    ! [Xa: $o] :
      ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))
    <=> (Xa) ) ).

% Max_singleton
tff(fact_1988_Max__singleton,axiom,
    ! [Xa: nat] : ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) = Xa ) ).

% Max_singleton
tff(fact_1989_Min__singleton,axiom,
    ! [Xa: int] : ( lattic8718645017227715691in_int(aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) = Xa ) ).

% Min_singleton
tff(fact_1990_Min__singleton,axiom,
    ! [Xa: $o] :
      ( aa_set_o_o(lattic1973801136483472281_Min_o,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))
    <=> (Xa) ) ).

% Min_singleton
tff(fact_1991_Min__singleton,axiom,
    ! [Xa: nat] : ( aa_set_nat_nat(lattic8721135487736765967in_nat,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) = Xa ) ).

% Min_singleton
tff(fact_1992_Max_Obounded__iff,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A4)),Xa)
        <=> ! [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X5),Xa) ) ) ) ) ).

% Max.bounded_iff
tff(fact_1993_Max_Obounded__iff,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A4)),(Xa))
        <=> ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X5)),(Xa)) ) ) ) ) ).

% Max.bounded_iff
tff(fact_1994_Max_Obounded__iff,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(A4)),Xa)
        <=> ! [X5: rat] :
              ( aa_set_rat_o(member_rat(X5),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Xa) ) ) ) ) ).

% Max.bounded_iff
tff(fact_1995_Max_Obounded__iff,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic4823215512031491691ax_num(A4)),Xa)
        <=> ! [X5: num] :
              ( aa_set_num_o(member_num(X5),A4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Xa) ) ) ) ) ).

% Max.bounded_iff
tff(fact_1996_Max_Obounded__iff,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A4)),Xa)
        <=> ! [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Xa) ) ) ) ) ).

% Max.bounded_iff
tff(fact_1997_Max_Obounded__iff,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)),Xa)
        <=> ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa) ) ) ) ) ).

% Max.bounded_iff
tff(fact_1998_Max__less__iff,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,lattic4901227151466704046nteger(A4)),Xa)
        <=> ! [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Xa) ) ) ) ) ).

% Max_less_iff
tff(fact_1999_Max__less__iff,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,aa_set_o_o(lattic1921953407002678535_Max_o,A4)),(Xa))
        <=> ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
             => aa_o_o(aa_o_fun_o_o(ord_less_o,(X5)),(Xa)) ) ) ) ) ).

% Max_less_iff
tff(fact_2000_Max__less__iff,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,lattic4275903605611617917x_real(A4)),Xa)
        <=> ! [X5: real] :
              ( aa_set_real_o(member_real(X5),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Xa) ) ) ) ) ).

% Max_less_iff
tff(fact_2001_Max__less__iff,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,lattic7630753665789217321ax_rat(A4)),Xa)
        <=> ! [X5: rat] :
              ( aa_set_rat_o(member_rat(X5),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Xa) ) ) ) ) ).

% Max_less_iff
tff(fact_2002_Max__less__iff,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,lattic4823215512031491691ax_num(A4)),Xa)
        <=> ! [X5: num] :
              ( aa_set_num_o(member_num(X5),A4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Xa) ) ) ) ) ).

% Max_less_iff
tff(fact_2003_Max__less__iff,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,lattic8263393255366662781ax_int(A4)),Xa)
        <=> ! [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Xa) ) ) ) ) ).

% Max_less_iff
tff(fact_2004_Max__less__iff,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)),Xa)
        <=> ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Xa) ) ) ) ) ).

% Max_less_iff
tff(fact_2005_Min_Obounded__iff,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),lattic1063845414844153500nteger(A4))
        <=> ! [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),X5) ) ) ) ) ).

% Min.bounded_iff
tff(fact_2006_Min_Obounded__iff,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),aa_set_o_o(lattic1973801136483472281_Min_o,A4))
        <=> ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(X5)) ) ) ) ) ).

% Min.bounded_iff
tff(fact_2007_Min_Obounded__iff,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic8086005427650270231in_rat(A4))
        <=> ! [X5: rat] :
              ( aa_set_rat_o(member_rat(X5),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X5) ) ) ) ) ).

% Min.bounded_iff
tff(fact_2008_Min_Obounded__iff,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),lattic5278467273892544601in_num(A4))
        <=> ! [X5: num] :
              ( aa_set_num_o(member_num(X5),A4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X5) ) ) ) ) ).

% Min.bounded_iff
tff(fact_2009_Min_Obounded__iff,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic8718645017227715691in_int(A4))
        <=> ! [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X5) ) ) ) ) ).

% Min.bounded_iff
tff(fact_2010_Min_Obounded__iff,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_set_nat_nat(lattic8721135487736765967in_nat,A4))
        <=> ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X5) ) ) ) ) ).

% Min.bounded_iff
tff(fact_2011_Min__gr__iff,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),lattic1063845414844153500nteger(A4))
        <=> ! [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),X5) ) ) ) ) ).

% Min_gr_iff
tff(fact_2012_Min__gr__iff,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa)),aa_set_o_o(lattic1973801136483472281_Min_o,A4))
        <=> ! [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
             => aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa)),(X5)) ) ) ) ) ).

% Min_gr_iff
tff(fact_2013_Min__gr__iff,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),lattic3629708407755379051n_real(A4))
        <=> ! [X5: real] :
              ( aa_set_real_o(member_real(X5),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X5) ) ) ) ) ).

% Min_gr_iff
tff(fact_2014_Min__gr__iff,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),lattic8086005427650270231in_rat(A4))
        <=> ! [X5: rat] :
              ( aa_set_rat_o(member_rat(X5),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X5) ) ) ) ) ).

% Min_gr_iff
tff(fact_2015_Min__gr__iff,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),lattic5278467273892544601in_num(A4))
        <=> ! [X5: num] :
              ( aa_set_num_o(member_num(X5),A4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X5) ) ) ) ) ).

% Min_gr_iff
tff(fact_2016_Min__gr__iff,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),lattic8718645017227715691in_int(A4))
        <=> ! [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X5) ) ) ) ) ).

% Min_gr_iff
tff(fact_2017_Min__gr__iff,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_set_nat_nat(lattic8721135487736765967in_nat,A4))
        <=> ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X5) ) ) ) ) ).

% Min_gr_iff
tff(fact_2018_Min_OcoboundedI,axiom,
    ! [A4: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(A),A4)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic3629708407755379051n_real(A4)),A) ) ) ).

% Min.coboundedI
tff(fact_2019_Min_OcoboundedI,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic1063845414844153500nteger(A4)),A) ) ) ).

% Min.coboundedI
tff(fact_2020_Min_OcoboundedI,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(A),A4)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(A4)),A) ) ) ).

% Min.coboundedI
tff(fact_2021_Min_OcoboundedI,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(A),A4)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic5278467273892544601in_num(A4)),A) ) ) ).

% Min.coboundedI
tff(fact_2022_Min_OcoboundedI,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(A4)),A) ) ) ).

% Min.coboundedI
tff(fact_2023_Min_OcoboundedI,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,A4)),A) ) ) ).

% Min.coboundedI
tff(fact_2024_Min__eqI,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [Y3: real] :
            ( aa_set_real_o(member_real(Y3),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y3) )
       => ( aa_set_real_o(member_real(Xa),A4)
         => ( lattic3629708407755379051n_real(A4) = Xa ) ) ) ) ).

% Min_eqI
tff(fact_2025_Min__eqI,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [Y3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(Y3),A4)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y3) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
         => ( lattic1063845414844153500nteger(A4) = Xa ) ) ) ) ).

% Min_eqI
tff(fact_2026_Min__eqI,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ! [Y3: rat] :
            ( aa_set_rat_o(member_rat(Y3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y3) )
       => ( aa_set_rat_o(member_rat(Xa),A4)
         => ( lattic8086005427650270231in_rat(A4) = Xa ) ) ) ) ).

% Min_eqI
tff(fact_2027_Min__eqI,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ! [Y3: num] :
            ( aa_set_num_o(member_num(Y3),A4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y3) )
       => ( aa_set_num_o(member_num(Xa),A4)
         => ( lattic5278467273892544601in_num(A4) = Xa ) ) ) ) ).

% Min_eqI
tff(fact_2028_Min__eqI,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [Y3: int] :
            ( aa_set_int_o(member_int(Y3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y3) )
       => ( aa_set_int_o(member_int(Xa),A4)
         => ( lattic8718645017227715691in_int(A4) = Xa ) ) ) ) ).

% Min_eqI
tff(fact_2029_Min__eqI,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y3) )
       => ( aa_set_nat_o(member_nat(Xa),A4)
         => ( aa_set_nat_nat(lattic8721135487736765967in_nat,A4) = Xa ) ) ) ) ).

% Min_eqI
tff(fact_2030_Min__le,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic3629708407755379051n_real(A4)),Xa) ) ) ).

% Min_le
tff(fact_2031_Min__le,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic1063845414844153500nteger(A4)),Xa) ) ) ).

% Min_le
tff(fact_2032_Min__le,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(Xa),A4)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(A4)),Xa) ) ) ).

% Min_le
tff(fact_2033_Min__le,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(Xa),A4)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic5278467273892544601in_num(A4)),Xa) ) ) ).

% Min_le
tff(fact_2034_Min__le,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(A4)),Xa) ) ) ).

% Min_le
tff(fact_2035_Min__le,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,A4)),Xa) ) ) ).

% Min_le
tff(fact_2036_Min__in,axiom,
    ! [A4: set_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => aa_set_real_o(member_real(lattic3629708407755379051n_real(A4)),A4) ) ) ).

% Min_in
tff(fact_2037_Min__in,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => aa_set5643387332196575111eger_o(member_Code_integer(lattic1063845414844153500nteger(A4)),A4) ) ) ).

% Min_in
tff(fact_2038_Min__in,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => aa_set_int_o(member_int(lattic8718645017227715691in_int(A4)),A4) ) ) ).

% Min_in
tff(fact_2039_Min__in,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => aa_set_o_o(member_o(aa_set_o_o(lattic1973801136483472281_Min_o,A4)),A4) ) ) ).

% Min_in
tff(fact_2040_Min__in,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(lattic8721135487736765967in_nat,A4)),A4) ) ) ).

% Min_in
tff(fact_2041_Max_OcoboundedI,axiom,
    ! [A4: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(A),A4)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),lattic4275903605611617917x_real(A4)) ) ) ).

% Max.coboundedI
tff(fact_2042_Max_OcoboundedI,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A),A4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),lattic4901227151466704046nteger(A4)) ) ) ).

% Max.coboundedI
tff(fact_2043_Max_OcoboundedI,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(A),A4)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),lattic7630753665789217321ax_rat(A4)) ) ) ).

% Max.coboundedI
tff(fact_2044_Max_OcoboundedI,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(A),A4)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),lattic4823215512031491691ax_num(A4)) ) ) ).

% Max.coboundedI
tff(fact_2045_Max_OcoboundedI,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(A),A4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),lattic8263393255366662781ax_int(A4)) ) ) ).

% Max.coboundedI
tff(fact_2046_Max_OcoboundedI,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(A),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)) ) ) ).

% Max.coboundedI
tff(fact_2047_Max__eq__if,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
             => ? [Xa2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),B4)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa2) ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),B4)
               => ? [Xa2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),A4)
                    & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa2) ) )
           => ( lattic4901227151466704046nteger(A4) = lattic4901227151466704046nteger(B4) ) ) ) ) ) ).

% Max_eq_if
tff(fact_2048_Max__eq__if,axiom,
    ! [A4: set_rat,B4: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(finite_finite_rat,B4)
       => ( ! [X: rat] :
              ( aa_set_rat_o(member_rat(X),A4)
             => ? [Xa2: rat] :
                  ( aa_set_rat_o(member_rat(Xa2),B4)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa2) ) )
         => ( ! [X: rat] :
                ( aa_set_rat_o(member_rat(X),B4)
               => ? [Xa2: rat] :
                    ( aa_set_rat_o(member_rat(Xa2),A4)
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa2) ) )
           => ( lattic7630753665789217321ax_rat(A4) = lattic7630753665789217321ax_rat(B4) ) ) ) ) ) ).

% Max_eq_if
tff(fact_2049_Max__eq__if,axiom,
    ! [A4: set_num,B4: set_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(finite_finite_num,B4)
       => ( ! [X: num] :
              ( aa_set_num_o(member_num(X),A4)
             => ? [Xa2: num] :
                  ( aa_set_num_o(member_num(Xa2),B4)
                  & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa2) ) )
         => ( ! [X: num] :
                ( aa_set_num_o(member_num(X),B4)
               => ? [Xa2: num] :
                    ( aa_set_num_o(member_num(Xa2),A4)
                    & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa2) ) )
           => ( lattic4823215512031491691ax_num(A4) = lattic4823215512031491691ax_num(B4) ) ) ) ) ) ).

% Max_eq_if
tff(fact_2050_Max__eq__if,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),A4)
             => ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),B4)
                  & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2) ) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),B4)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),A4)
                    & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2) ) )
           => ( lattic8263393255366662781ax_int(A4) = lattic8263393255366662781ax_int(B4) ) ) ) ) ) ).

% Max_eq_if
tff(fact_2051_Max__eq__if,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
             => ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa2) ) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),B4)
               => ? [Xa2: nat] :
                    ( aa_set_nat_o(member_nat(Xa2),A4)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa2) ) )
           => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A4) = aa_set_nat_nat(lattic8265883725875713057ax_nat,B4) ) ) ) ) ) ).

% Max_eq_if
tff(fact_2052_Max__eqI,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [Y3: real] :
            ( aa_set_real_o(member_real(Y3),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Xa) )
       => ( aa_set_real_o(member_real(Xa),A4)
         => ( lattic4275903605611617917x_real(A4) = Xa ) ) ) ) ).

% Max_eqI
tff(fact_2053_Max__eqI,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [Y3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(Y3),A4)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y3),Xa) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
         => ( lattic4901227151466704046nteger(A4) = Xa ) ) ) ) ).

% Max_eqI
tff(fact_2054_Max__eqI,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ! [Y3: rat] :
            ( aa_set_rat_o(member_rat(Y3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y3),Xa) )
       => ( aa_set_rat_o(member_rat(Xa),A4)
         => ( lattic7630753665789217321ax_rat(A4) = Xa ) ) ) ) ).

% Max_eqI
tff(fact_2055_Max__eqI,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ! [Y3: num] :
            ( aa_set_num_o(member_num(Y3),A4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y3),Xa) )
       => ( aa_set_num_o(member_num(Xa),A4)
         => ( lattic4823215512031491691ax_num(A4) = Xa ) ) ) ) ).

% Max_eqI
tff(fact_2056_Max__eqI,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [Y3: int] :
            ( aa_set_int_o(member_int(Y3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y3),Xa) )
       => ( aa_set_int_o(member_int(Xa),A4)
         => ( lattic8263393255366662781ax_int(A4) = Xa ) ) ) ) ).

% Max_eqI
tff(fact_2057_Max__eqI,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Xa) )
       => ( aa_set_nat_o(member_nat(Xa),A4)
         => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A4) = Xa ) ) ) ) ).

% Max_eqI
tff(fact_2058_Max__ge,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),lattic4275903605611617917x_real(A4)) ) ) ).

% Max_ge
tff(fact_2059_Max__ge,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),lattic4901227151466704046nteger(A4)) ) ) ).

% Max_ge
tff(fact_2060_Max__ge,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( aa_set_rat_o(member_rat(Xa),A4)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic7630753665789217321ax_rat(A4)) ) ) ).

% Max_ge
tff(fact_2061_Max__ge,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( aa_set_num_o(member_num(Xa),A4)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),lattic4823215512031491691ax_num(A4)) ) ) ).

% Max_ge
tff(fact_2062_Max__ge,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic8263393255366662781ax_int(A4)) ) ) ).

% Max_ge
tff(fact_2063_Max__ge,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)) ) ) ).

% Max_ge
tff(fact_2064_Max__in,axiom,
    ! [A4: set_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => aa_set_real_o(member_real(lattic4275903605611617917x_real(A4)),A4) ) ) ).

% Max_in
tff(fact_2065_Max__in,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => aa_set5643387332196575111eger_o(member_Code_integer(lattic4901227151466704046nteger(A4)),A4) ) ) ).

% Max_in
tff(fact_2066_Max__in,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => aa_set_int_o(member_int(lattic8263393255366662781ax_int(A4)),A4) ) ) ).

% Max_in
tff(fact_2067_Max__in,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => aa_set_o_o(member_o(aa_set_o_o(lattic1921953407002678535_Max_o,A4)),A4) ) ) ).

% Max_in
tff(fact_2068_Max__in,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)),A4) ) ) ).

% Max_in
tff(fact_2069_Min__eq__iff,axiom,
    ! [A4: set_real,M: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ( lattic3629708407755379051n_real(A4) = M )
        <=> ( aa_set_real_o(member_real(M),A4)
            & ! [X5: real] :
                ( aa_set_real_o(member_real(X5),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M),X5) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2070_Min__eq__iff,axiom,
    ! [A4: set_Code_integer,M: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( lattic1063845414844153500nteger(A4) = M )
        <=> ( aa_set5643387332196575111eger_o(member_Code_integer(M),A4)
            & ! [X5: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),X5) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2071_Min__eq__iff,axiom,
    ! [A4: set_o,M: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ( aa_set_o_o(lattic1973801136483472281_Min_o,A4)
          <=> (M) )
        <=> ( aa_set_o_o(member_o((M)),A4)
            & ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(M)),(X5)) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2072_Min__eq__iff,axiom,
    ! [A4: set_rat,M: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( ( lattic8086005427650270231in_rat(A4) = M )
        <=> ( aa_set_rat_o(member_rat(M),A4)
            & ! [X5: rat] :
                ( aa_set_rat_o(member_rat(X5),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),X5) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2073_Min__eq__iff,axiom,
    ! [A4: set_num,M: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( ( lattic5278467273892544601in_num(A4) = M )
        <=> ( aa_set_num_o(member_num(M),A4)
            & ! [X5: num] :
                ( aa_set_num_o(member_num(X5),A4)
               => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),X5) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2074_Min__eq__iff,axiom,
    ! [A4: set_int,M: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ( lattic8718645017227715691in_int(A4) = M )
        <=> ( aa_set_int_o(member_int(M),A4)
            & ! [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),X5) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2075_Min__eq__iff,axiom,
    ! [A4: set_nat,M: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ( aa_set_nat_nat(lattic8721135487736765967in_nat,A4) = M )
        <=> ( aa_set_nat_o(member_nat(M),A4)
            & ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),X5) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_2076_Min__le__iff,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic1063845414844153500nteger(A4)),Xa)
        <=> ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X5),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_2077_Min__le__iff,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1973801136483472281_Min_o,A4)),(Xa))
        <=> ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
              & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X5)),(Xa)) ) ) ) ) ).

% Min_le_iff
tff(fact_2078_Min__le__iff,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(A4)),Xa)
        <=> ? [X5: rat] :
              ( aa_set_rat_o(member_rat(X5),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_2079_Min__le__iff,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic5278467273892544601in_num(A4)),Xa)
        <=> ? [X5: num] :
              ( aa_set_num_o(member_num(X5),A4)
              & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_2080_Min__le__iff,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(A4)),Xa)
        <=> ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_2081_Min__le__iff,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,A4)),Xa)
        <=> ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Xa) ) ) ) ) ).

% Min_le_iff
tff(fact_2082_eq__Min__iff,axiom,
    ! [A4: set_real,M: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ( M = lattic3629708407755379051n_real(A4) )
        <=> ( aa_set_real_o(member_real(M),A4)
            & ! [X5: real] :
                ( aa_set_real_o(member_real(X5),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M),X5) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2083_eq__Min__iff,axiom,
    ! [A4: set_Code_integer,M: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( M = lattic1063845414844153500nteger(A4) )
        <=> ( aa_set5643387332196575111eger_o(member_Code_integer(M),A4)
            & ! [X5: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),X5) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2084_eq__Min__iff,axiom,
    ! [A4: set_o,M: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ( (M)
          <=> aa_set_o_o(lattic1973801136483472281_Min_o,A4) )
        <=> ( aa_set_o_o(member_o((M)),A4)
            & ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(M)),(X5)) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2085_eq__Min__iff,axiom,
    ! [A4: set_rat,M: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( ( M = lattic8086005427650270231in_rat(A4) )
        <=> ( aa_set_rat_o(member_rat(M),A4)
            & ! [X5: rat] :
                ( aa_set_rat_o(member_rat(X5),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),X5) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2086_eq__Min__iff,axiom,
    ! [A4: set_num,M: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( ( M = lattic5278467273892544601in_num(A4) )
        <=> ( aa_set_num_o(member_num(M),A4)
            & ! [X5: num] :
                ( aa_set_num_o(member_num(X5),A4)
               => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),X5) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2087_eq__Min__iff,axiom,
    ! [A4: set_int,M: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ( M = lattic8718645017227715691in_int(A4) )
        <=> ( aa_set_int_o(member_int(M),A4)
            & ! [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),X5) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2088_eq__Min__iff,axiom,
    ! [A4: set_nat,M: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ( M = aa_set_nat_nat(lattic8721135487736765967in_nat,A4) )
        <=> ( aa_set_nat_o(member_nat(M),A4)
            & ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),X5) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_2089_Min_OboundedE,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),lattic3629708407755379051n_real(A4))
         => ! [A8: real] :
              ( aa_set_real_o(member_real(A8),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A8) ) ) ) ) ).

% Min.boundedE
tff(fact_2090_Min_OboundedE,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),lattic1063845414844153500nteger(A4))
         => ! [A8: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A8),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),A8) ) ) ) ) ).

% Min.boundedE
tff(fact_2091_Min_OboundedE,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),aa_set_o_o(lattic1973801136483472281_Min_o,A4))
         => ! [A8: $o] :
              ( aa_set_o_o(member_o((A8)),A4)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(A8)) ) ) ) ) ).

% Min.boundedE
tff(fact_2092_Min_OboundedE,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic8086005427650270231in_rat(A4))
         => ! [A8: rat] :
              ( aa_set_rat_o(member_rat(A8),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A8) ) ) ) ) ).

% Min.boundedE
tff(fact_2093_Min_OboundedE,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),lattic5278467273892544601in_num(A4))
         => ! [A8: num] :
              ( aa_set_num_o(member_num(A8),A4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),A8) ) ) ) ) ).

% Min.boundedE
tff(fact_2094_Min_OboundedE,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic8718645017227715691in_int(A4))
         => ! [A8: int] :
              ( aa_set_int_o(member_int(A8),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A8) ) ) ) ) ).

% Min.boundedE
tff(fact_2095_Min_OboundedE,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_set_nat_nat(lattic8721135487736765967in_nat,A4))
         => ! [A8: nat] :
              ( aa_set_nat_o(member_nat(A8),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A8) ) ) ) ) ).

% Min.boundedE
tff(fact_2096_Min_OboundedI,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A3) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),lattic3629708407755379051n_real(A4)) ) ) ) ).

% Min.boundedI
tff(fact_2097_Min_OboundedI,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [A3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),A3) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),lattic1063845414844153500nteger(A4)) ) ) ) ).

% Min.boundedI
tff(fact_2098_Min_OboundedI,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A4)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(A3)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),aa_set_o_o(lattic1973801136483472281_Min_o,A4)) ) ) ) ).

% Min.boundedI
tff(fact_2099_Min_OboundedI,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( aa_set_rat_o(member_rat(A3),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A3) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic8086005427650270231in_rat(A4)) ) ) ) ).

% Min.boundedI
tff(fact_2100_Min_OboundedI,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( ! [A3: num] :
              ( aa_set_num_o(member_num(A3),A4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),A3) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),lattic5278467273892544601in_num(A4)) ) ) ) ).

% Min.boundedI
tff(fact_2101_Min_OboundedI,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A3) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic8718645017227715691in_int(A4)) ) ) ) ).

% Min.boundedI
tff(fact_2102_Min_OboundedI,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),A3) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_set_nat_nat(lattic8721135487736765967in_nat,A4)) ) ) ) ).

% Min.boundedI
tff(fact_2103_Min__less__iff,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,lattic1063845414844153500nteger(A4)),Xa)
        <=> ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X5),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_2104_Min__less__iff,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,aa_set_o_o(lattic1973801136483472281_Min_o,A4)),(Xa))
        <=> ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
              & aa_o_o(aa_o_fun_o_o(ord_less_o,(X5)),(Xa)) ) ) ) ) ).

% Min_less_iff
tff(fact_2105_Min__less__iff,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,lattic3629708407755379051n_real(A4)),Xa)
        <=> ? [X5: real] :
              ( aa_set_real_o(member_real(X5),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,X5),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_2106_Min__less__iff,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,lattic8086005427650270231in_rat(A4)),Xa)
        <=> ? [X5: rat] :
              ( aa_set_rat_o(member_rat(X5),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X5),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_2107_Min__less__iff,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,lattic5278467273892544601in_num(A4)),Xa)
        <=> ? [X5: num] :
              ( aa_set_num_o(member_num(X5),A4)
              & aa_num_o(aa_num_fun_num_o(ord_less_num,X5),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_2108_Min__less__iff,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,lattic8718645017227715691in_int(A4)),Xa)
        <=> ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,X5),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_2109_Min__less__iff,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,A4)),Xa)
        <=> ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Xa) ) ) ) ) ).

% Min_less_iff
tff(fact_2110_Min__insert2,axiom,
    ! [A4: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [B3: $o] :
            ( aa_set_o_o(member_o((B3)),A4)
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A)),(B3)) )
       => ( aa_set_o_o(lattic1973801136483472281_Min_o,aa_set_o_set_o(insert_o((A)),A4))
        <=> (A) ) ) ) ).

% Min_insert2
tff(fact_2111_Min__insert2,axiom,
    ! [A4: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [B3: real] :
            ( aa_set_real_o(member_real(B3),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B3) )
       => ( lattic3629708407755379051n_real(aa_set_real_set_real(insert_real(A),A4)) = A ) ) ) ).

% Min_insert2
tff(fact_2112_Min__insert2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [B3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(B3),A4)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B3) )
       => ( lattic1063845414844153500nteger(insert_Code_integer(A,A4)) = A ) ) ) ).

% Min_insert2
tff(fact_2113_Min__insert2,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ! [B3: rat] :
            ( aa_set_rat_o(member_rat(B3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B3) )
       => ( lattic8086005427650270231in_rat(insert_rat(A,A4)) = A ) ) ) ).

% Min_insert2
tff(fact_2114_Min__insert2,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ! [B3: num] :
            ( aa_set_num_o(member_num(B3),A4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B3) )
       => ( lattic5278467273892544601in_num(insert_num(A,A4)) = A ) ) ) ).

% Min_insert2
tff(fact_2115_Min__insert2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [B3: int] :
            ( aa_set_int_o(member_int(B3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B3) )
       => ( lattic8718645017227715691in_int(aa_set_int_set_int(insert_int(A),A4)) = A ) ) ) ).

% Min_insert2
tff(fact_2116_Min__insert2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [B3: nat] :
            ( aa_set_nat_o(member_nat(B3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B3) )
       => ( aa_set_nat_nat(lattic8721135487736765967in_nat,aa_set_nat_set_nat(insert_nat(A),A4)) = A ) ) ) ).

% Min_insert2
tff(fact_2117_Max__eq__iff,axiom,
    ! [A4: set_real,M: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ( lattic4275903605611617917x_real(A4) = M )
        <=> ( aa_set_real_o(member_real(M),A4)
            & ! [X5: real] :
                ( aa_set_real_o(member_real(X5),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),M) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2118_Max__eq__iff,axiom,
    ! [A4: set_Code_integer,M: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( lattic4901227151466704046nteger(A4) = M )
        <=> ( aa_set5643387332196575111eger_o(member_Code_integer(M),A4)
            & ! [X5: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X5),M) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2119_Max__eq__iff,axiom,
    ! [A4: set_o,M: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ( aa_set_o_o(lattic1921953407002678535_Max_o,A4)
          <=> (M) )
        <=> ( aa_set_o_o(member_o((M)),A4)
            & ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X5)),(M)) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2120_Max__eq__iff,axiom,
    ! [A4: set_rat,M: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( ( lattic7630753665789217321ax_rat(A4) = M )
        <=> ( aa_set_rat_o(member_rat(M),A4)
            & ! [X5: rat] :
                ( aa_set_rat_o(member_rat(X5),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),M) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2121_Max__eq__iff,axiom,
    ! [A4: set_num,M: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( ( lattic4823215512031491691ax_num(A4) = M )
        <=> ( aa_set_num_o(member_num(M),A4)
            & ! [X5: num] :
                ( aa_set_num_o(member_num(X5),A4)
               => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),M) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2122_Max__eq__iff,axiom,
    ! [A4: set_int,M: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ( lattic8263393255366662781ax_int(A4) = M )
        <=> ( aa_set_int_o(member_int(M),A4)
            & ! [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),M) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2123_Max__eq__iff,axiom,
    ! [A4: set_nat,M: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A4) = M )
        <=> ( aa_set_nat_o(member_nat(M),A4)
            & ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),M) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_2124_Max__ge__iff,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),lattic4901227151466704046nteger(A4))
        <=> ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),X5) ) ) ) ) ).

% Max_ge_iff
tff(fact_2125_Max__ge__iff,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),aa_set_o_o(lattic1921953407002678535_Max_o,A4))
        <=> ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
              & aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(X5)) ) ) ) ) ).

% Max_ge_iff
tff(fact_2126_Max__ge__iff,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),lattic7630753665789217321ax_rat(A4))
        <=> ? [X5: rat] :
              ( aa_set_rat_o(member_rat(X5),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X5) ) ) ) ) ).

% Max_ge_iff
tff(fact_2127_Max__ge__iff,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),lattic4823215512031491691ax_num(A4))
        <=> ? [X5: num] :
              ( aa_set_num_o(member_num(X5),A4)
              & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X5) ) ) ) ) ).

% Max_ge_iff
tff(fact_2128_Max__ge__iff,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),lattic8263393255366662781ax_int(A4))
        <=> ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X5) ) ) ) ) ).

% Max_ge_iff
tff(fact_2129_Max__ge__iff,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_set_nat_nat(lattic8265883725875713057ax_nat,A4))
        <=> ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X5) ) ) ) ) ).

% Max_ge_iff
tff(fact_2130_eq__Max__iff,axiom,
    ! [A4: set_real,M: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ( M = lattic4275903605611617917x_real(A4) )
        <=> ( aa_set_real_o(member_real(M),A4)
            & ! [X5: real] :
                ( aa_set_real_o(member_real(X5),A4)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),M) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2131_eq__Max__iff,axiom,
    ! [A4: set_Code_integer,M: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ( M = lattic4901227151466704046nteger(A4) )
        <=> ( aa_set5643387332196575111eger_o(member_Code_integer(M),A4)
            & ! [X5: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
               => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X5),M) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2132_eq__Max__iff,axiom,
    ! [A4: set_o,M: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ( (M)
          <=> aa_set_o_o(lattic1921953407002678535_Max_o,A4) )
        <=> ( aa_set_o_o(member_o((M)),A4)
            & ! [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
               => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(X5)),(M)) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2133_eq__Max__iff,axiom,
    ! [A4: set_rat,M: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( ( M = lattic7630753665789217321ax_rat(A4) )
        <=> ( aa_set_rat_o(member_rat(M),A4)
            & ! [X5: rat] :
                ( aa_set_rat_o(member_rat(X5),A4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X5),M) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2134_eq__Max__iff,axiom,
    ! [A4: set_num,M: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( ( M = lattic4823215512031491691ax_num(A4) )
        <=> ( aa_set_num_o(member_num(M),A4)
            & ! [X5: num] :
                ( aa_set_num_o(member_num(X5),A4)
               => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X5),M) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2135_eq__Max__iff,axiom,
    ! [A4: set_int,M: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ( M = lattic8263393255366662781ax_int(A4) )
        <=> ( aa_set_int_o(member_int(M),A4)
            & ! [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),M) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2136_eq__Max__iff,axiom,
    ! [A4: set_nat,M: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ( M = aa_set_nat_nat(lattic8265883725875713057ax_nat,A4) )
        <=> ( aa_set_nat_o(member_nat(M),A4)
            & ! [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),M) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_2137_Max_OboundedE,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic4275903605611617917x_real(A4)),Xa)
         => ! [A8: real] :
              ( aa_set_real_o(member_real(A8),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A8),Xa) ) ) ) ) ).

% Max.boundedE
tff(fact_2138_Max_OboundedE,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A4)),Xa)
         => ! [A8: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A8),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A8),Xa) ) ) ) ) ).

% Max.boundedE
tff(fact_2139_Max_OboundedE,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A4)),(Xa))
         => ! [A8: $o] :
              ( aa_set_o_o(member_o((A8)),A4)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A8)),(Xa)) ) ) ) ) ).

% Max.boundedE
tff(fact_2140_Max_OboundedE,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(A4)),Xa)
         => ! [A8: rat] :
              ( aa_set_rat_o(member_rat(A8),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A8),Xa) ) ) ) ) ).

% Max.boundedE
tff(fact_2141_Max_OboundedE,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic4823215512031491691ax_num(A4)),Xa)
         => ! [A8: num] :
              ( aa_set_num_o(member_num(A8),A4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A8),Xa) ) ) ) ) ).

% Max.boundedE
tff(fact_2142_Max_OboundedE,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A4)),Xa)
         => ! [A8: int] :
              ( aa_set_int_o(member_int(A8),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A8),Xa) ) ) ) ) ).

% Max.boundedE
tff(fact_2143_Max_OboundedE,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)),Xa)
         => ! [A8: nat] :
              ( aa_set_nat_o(member_nat(A8),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A8),Xa) ) ) ) ) ).

% Max.boundedE
tff(fact_2144_Max_OboundedI,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),Xa) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,lattic4275903605611617917x_real(A4)),Xa) ) ) ) ).

% Max.boundedI
tff(fact_2145_Max_OboundedI,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [A3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A4)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A3),Xa) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A4)),Xa) ) ) ) ).

% Max.boundedI
tff(fact_2146_Max_OboundedI,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ! [A3: $o] :
              ( aa_set_o_o(member_o((A3)),A4)
             => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(A3)),(Xa)) )
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A4)),(Xa)) ) ) ) ).

% Max.boundedI
tff(fact_2147_Max_OboundedI,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( ! [A3: rat] :
              ( aa_set_rat_o(member_rat(A3),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),Xa) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(A4)),Xa) ) ) ) ).

% Max.boundedI
tff(fact_2148_Max_OboundedI,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( ! [A3: num] :
              ( aa_set_num_o(member_num(A3),A4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A3),Xa) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic4823215512031491691ax_num(A4)),Xa) ) ) ) ).

% Max.boundedI
tff(fact_2149_Max_OboundedI,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [A3: int] :
              ( aa_set_int_o(member_int(A3),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),Xa) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A4)),Xa) ) ) ) ).

% Max.boundedI
tff(fact_2150_Max_OboundedI,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [A3: nat] :
              ( aa_set_nat_o(member_nat(A3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),Xa) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)),Xa) ) ) ) ).

% Max.boundedI
tff(fact_2151_Max__gr__iff,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),lattic4901227151466704046nteger(A4))
        <=> ? [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
              & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),X5) ) ) ) ) ).

% Max_gr_iff
tff(fact_2152_Max__gr__iff,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa)),aa_set_o_o(lattic1921953407002678535_Max_o,A4))
        <=> ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),A4)
              & aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa)),(X5)) ) ) ) ) ).

% Max_gr_iff
tff(fact_2153_Max__gr__iff,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),lattic4275903605611617917x_real(A4))
        <=> ? [X5: real] :
              ( aa_set_real_o(member_real(X5),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X5) ) ) ) ) ).

% Max_gr_iff
tff(fact_2154_Max__gr__iff,axiom,
    ! [A4: set_rat,Xa: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),lattic7630753665789217321ax_rat(A4))
        <=> ? [X5: rat] :
              ( aa_set_rat_o(member_rat(X5),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X5) ) ) ) ) ).

% Max_gr_iff
tff(fact_2155_Max__gr__iff,axiom,
    ! [A4: set_num,Xa: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),lattic4823215512031491691ax_num(A4))
        <=> ? [X5: num] :
              ( aa_set_num_o(member_num(X5),A4)
              & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X5) ) ) ) ) ).

% Max_gr_iff
tff(fact_2156_Max__gr__iff,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),lattic8263393255366662781ax_int(A4))
        <=> ? [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X5) ) ) ) ) ).

% Max_gr_iff
tff(fact_2157_Max__gr__iff,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_set_nat_nat(lattic8265883725875713057ax_nat,A4))
        <=> ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X5) ) ) ) ) ).

% Max_gr_iff
tff(fact_2158_Max__insert2,axiom,
    ! [A4: set_o,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [B3: $o] :
            ( aa_set_o_o(member_o((B3)),A4)
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B3)),(A)) )
       => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(insert_o((A)),A4))
        <=> (A) ) ) ) ).

% Max_insert2
tff(fact_2159_Max__insert2,axiom,
    ! [A4: set_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [B3: real] :
            ( aa_set_real_o(member_real(B3),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),A) )
       => ( lattic4275903605611617917x_real(aa_set_real_set_real(insert_real(A),A4)) = A ) ) ) ).

% Max_insert2
tff(fact_2160_Max__insert2,axiom,
    ! [A4: set_Code_integer,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [B3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(B3),A4)
           => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B3),A) )
       => ( lattic4901227151466704046nteger(insert_Code_integer(A,A4)) = A ) ) ) ).

% Max_insert2
tff(fact_2161_Max__insert2,axiom,
    ! [A4: set_rat,A: rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ! [B3: rat] :
            ( aa_set_rat_o(member_rat(B3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B3),A) )
       => ( lattic7630753665789217321ax_rat(insert_rat(A,A4)) = A ) ) ) ).

% Max_insert2
tff(fact_2162_Max__insert2,axiom,
    ! [A4: set_num,A: num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ! [B3: num] :
            ( aa_set_num_o(member_num(B3),A4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B3),A) )
       => ( lattic4823215512031491691ax_num(insert_num(A,A4)) = A ) ) ) ).

% Max_insert2
tff(fact_2163_Max__insert2,axiom,
    ! [A4: set_int,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [B3: int] :
            ( aa_set_int_o(member_int(B3),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),A) )
       => ( lattic8263393255366662781ax_int(aa_set_int_set_int(insert_int(A),A4)) = A ) ) ) ).

% Max_insert2
tff(fact_2164_Max__insert2,axiom,
    ! [A4: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [B3: nat] :
            ( aa_set_nat_o(member_nat(B3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B3),A) )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(A),A4)) = A ) ) ) ).

% Max_insert2
tff(fact_2165_Min__antimono,axiom,
    ! [M7: set_Code_integer,N6: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(M7),N6)
     => ( ( M7 != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,N6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic1063845414844153500nteger(N6)),lattic1063845414844153500nteger(M7)) ) ) ) ).

% Min_antimono
tff(fact_2166_Min__antimono,axiom,
    ! [M7: set_o,N6: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(M7),N6)
     => ( ( M7 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,N6)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1973801136483472281_Min_o,N6)),aa_set_o_o(lattic1973801136483472281_Min_o,M7)) ) ) ) ).

% Min_antimono
tff(fact_2167_Min__antimono,axiom,
    ! [M7: set_rat,N6: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(M7),N6)
     => ( ( M7 != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,N6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(N6)),lattic8086005427650270231in_rat(M7)) ) ) ) ).

% Min_antimono
tff(fact_2168_Min__antimono,axiom,
    ! [M7: set_num,N6: set_num] :
      ( aa_set_num_o(ord_less_eq_set_num(M7),N6)
     => ( ( M7 != bot_bot_set_num )
       => ( aa_set_num_o(finite_finite_num,N6)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic5278467273892544601in_num(N6)),lattic5278467273892544601in_num(M7)) ) ) ) ).

% Min_antimono
tff(fact_2169_Min__antimono,axiom,
    ! [M7: set_int,N6: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(M7),N6)
     => ( ( M7 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,N6)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(N6)),lattic8718645017227715691in_int(M7)) ) ) ) ).

% Min_antimono
tff(fact_2170_Min__antimono,axiom,
    ! [M7: set_nat,N6: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,M7),N6)
     => ( ( M7 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,N6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,N6)),aa_set_nat_nat(lattic8721135487736765967in_nat,M7)) ) ) ) ).

% Min_antimono
tff(fact_2171_Min_Osubset__imp,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic1063845414844153500nteger(B4)),lattic1063845414844153500nteger(A4)) ) ) ) ).

% Min.subset_imp
tff(fact_2172_Min_Osubset__imp,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1973801136483472281_Min_o,B4)),aa_set_o_o(lattic1973801136483472281_Min_o,A4)) ) ) ) ).

% Min.subset_imp
tff(fact_2173_Min_Osubset__imp,axiom,
    ! [A4: set_rat,B4: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(A4),B4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,B4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic8086005427650270231in_rat(B4)),lattic8086005427650270231in_rat(A4)) ) ) ) ).

% Min.subset_imp
tff(fact_2174_Min_Osubset__imp,axiom,
    ! [A4: set_num,B4: set_num] :
      ( aa_set_num_o(ord_less_eq_set_num(A4),B4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_set_num_o(finite_finite_num,B4)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic5278467273892544601in_num(B4)),lattic5278467273892544601in_num(A4)) ) ) ) ).

% Min.subset_imp
tff(fact_2175_Min_Osubset__imp,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8718645017227715691in_int(B4)),lattic8718645017227715691in_int(A4)) ) ) ) ).

% Min.subset_imp
tff(fact_2176_Min_Osubset__imp,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8721135487736765967in_nat,B4)),aa_set_nat_nat(lattic8721135487736765967in_nat,A4)) ) ) ) ).

% Min.subset_imp
tff(fact_2177_Max__mono,axiom,
    ! [M7: set_Code_integer,N6: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(M7),N6)
     => ( ( M7 != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,N6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(M7)),lattic4901227151466704046nteger(N6)) ) ) ) ).

% Max_mono
tff(fact_2178_Max__mono,axiom,
    ! [M7: set_o,N6: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(M7),N6)
     => ( ( M7 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,N6)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,M7)),aa_set_o_o(lattic1921953407002678535_Max_o,N6)) ) ) ) ).

% Max_mono
tff(fact_2179_Max__mono,axiom,
    ! [M7: set_rat,N6: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(M7),N6)
     => ( ( M7 != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,N6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(M7)),lattic7630753665789217321ax_rat(N6)) ) ) ) ).

% Max_mono
tff(fact_2180_Max__mono,axiom,
    ! [M7: set_num,N6: set_num] :
      ( aa_set_num_o(ord_less_eq_set_num(M7),N6)
     => ( ( M7 != bot_bot_set_num )
       => ( aa_set_num_o(finite_finite_num,N6)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic4823215512031491691ax_num(M7)),lattic4823215512031491691ax_num(N6)) ) ) ) ).

% Max_mono
tff(fact_2181_Max__mono,axiom,
    ! [M7: set_int,N6: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(M7),N6)
     => ( ( M7 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,N6)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(M7)),lattic8263393255366662781ax_int(N6)) ) ) ) ).

% Max_mono
tff(fact_2182_Max__mono,axiom,
    ! [M7: set_nat,N6: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,M7),N6)
     => ( ( M7 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,N6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,M7)),aa_set_nat_nat(lattic8265883725875713057ax_nat,N6)) ) ) ) ).

% Max_mono
tff(fact_2183_Max_Osubset__imp,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,lattic4901227151466704046nteger(A4)),lattic4901227151466704046nteger(B4)) ) ) ) ).

% Max.subset_imp
tff(fact_2184_Max_Osubset__imp,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_set_o_o(finite_finite_o,B4)
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_set_o_o(lattic1921953407002678535_Max_o,A4)),aa_set_o_o(lattic1921953407002678535_Max_o,B4)) ) ) ) ).

% Max.subset_imp
tff(fact_2185_Max_Osubset__imp,axiom,
    ! [A4: set_rat,B4: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(A4),B4)
     => ( ( A4 != bot_bot_set_rat )
       => ( aa_set_rat_o(finite_finite_rat,B4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,lattic7630753665789217321ax_rat(A4)),lattic7630753665789217321ax_rat(B4)) ) ) ) ).

% Max.subset_imp
tff(fact_2186_Max_Osubset__imp,axiom,
    ! [A4: set_num,B4: set_num] :
      ( aa_set_num_o(ord_less_eq_set_num(A4),B4)
     => ( ( A4 != bot_bot_set_num )
       => ( aa_set_num_o(finite_finite_num,B4)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,lattic4823215512031491691ax_num(A4)),lattic4823215512031491691ax_num(B4)) ) ) ) ).

% Max.subset_imp
tff(fact_2187_Max_Osubset__imp,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_set_int_o(finite_finite_int,B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,lattic8263393255366662781ax_int(A4)),lattic8263393255366662781ax_int(B4)) ) ) ) ).

% Max.subset_imp
tff(fact_2188_Max_Osubset__imp,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,B4)) ) ) ) ).

% Max.subset_imp
tff(fact_2189_greater__shift,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xa)
    <=> vEBT_VEBT_greater(some_nat(Xa),some_nat(Y)) ) ).

% greater_shift
tff(fact_2190_less__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
    <=> vEBT_VEBT_less(some_nat(Xa),some_nat(Y)) ) ).

% less_shift
tff(fact_2191_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),Xa) = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_2192_the__elem__eq,axiom,
    ! [Xa: vEBT_VEBT] : ( the_elem_VEBT_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)) = Xa ) ).

% the_elem_eq
tff(fact_2193_the__elem__eq,axiom,
    ! [Xa: nat] : ( aa_set_nat_nat(the_elem_nat,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) = Xa ) ).

% the_elem_eq
tff(fact_2194_the__elem__eq,axiom,
    ! [Xa: int] : ( the_elem_int(aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) = Xa ) ).

% the_elem_eq
tff(fact_2195_the__elem__eq,axiom,
    ! [Xa: $o] :
      ( aa_set_o_o(the_elem_o,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))
    <=> (Xa) ) ).

% the_elem_eq
tff(fact_2196_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xa) = Y )
     => ( ( ( Xa = vEBT_Leaf($false,$false) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv: $o] : ( Xa = vEBT_Leaf($true,(Uv)) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu: $o] : ( Xa = vEBT_Leaf((Uu),$true) )
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy) )
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) )
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_2197_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Vi2) = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_2198_is__singletonI,axiom,
    ! [Xa: vEBT_VEBT] : aa_set_VEBT_VEBT_o(is_sin24926331636114728T_VEBT,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)) ).

% is_singletonI
tff(fact_2199_is__singletonI,axiom,
    ! [Xa: nat] : aa_set_nat_o(is_singleton_nat,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) ).

% is_singletonI
tff(fact_2200_is__singletonI,axiom,
    ! [Xa: int] : aa_set_int_o(is_singleton_int,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) ).

% is_singletonI
tff(fact_2201_is__singletonI,axiom,
    ! [Xa: $o] : aa_set_o_o(is_singleton_o,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)) ).

% is_singletonI
tff(fact_2202_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Vi2) = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_2203_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Vj2) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_2204_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Vj2) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_2205_is__singleton__the__elem,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(is_sin24926331636114728T_VEBT,A4)
    <=> ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(the_elem_VEBT_VEBT(A4)),bot_bo8194388402131092736T_VEBT) ) ) ).

% is_singleton_the_elem
tff(fact_2206_is__singleton__the__elem,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
    <=> ( A4 = aa_set_nat_set_nat(insert_nat(aa_set_nat_nat(the_elem_nat,A4)),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_2207_is__singleton__the__elem,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_singleton_int,A4)
    <=> ( A4 = aa_set_int_set_int(insert_int(the_elem_int(A4)),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_2208_is__singleton__the__elem,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_singleton_o,A4)
    <=> ( A4 = aa_set_o_set_o(insert_o(aa_set_o_o(the_elem_o,A4)),bot_bot_set_o) ) ) ).

% is_singleton_the_elem
tff(fact_2209_is__singletonI_H,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( A4 != bot_bo8194388402131092736T_VEBT )
     => ( ! [X: vEBT_VEBT,Y3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
           => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y3),A4)
             => ( X = Y3 ) ) )
       => aa_set_VEBT_VEBT_o(is_sin24926331636114728T_VEBT,A4) ) ) ).

% is_singletonI'
tff(fact_2210_is__singletonI_H,axiom,
    ! [A4: set_real] :
      ( ( A4 != bot_bot_set_real )
     => ( ! [X: real,Y3: real] :
            ( aa_set_real_o(member_real(X),A4)
           => ( aa_set_real_o(member_real(Y3),A4)
             => ( X = Y3 ) ) )
       => aa_set_real_o(is_singleton_real,A4) ) ) ).

% is_singletonI'
tff(fact_2211_is__singletonI_H,axiom,
    ! [A4: set_set_nat] :
      ( ( A4 != bot_bot_set_set_nat )
     => ( ! [X: set_nat,Y3: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X),A4)
           => ( aa_set_set_nat_o(member_set_nat(Y3),A4)
             => ( X = Y3 ) ) )
       => aa_set_set_nat_o(is_singleton_set_nat,A4) ) ) ).

% is_singletonI'
tff(fact_2212_is__singletonI_H,axiom,
    ! [A4: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X: nat,Y3: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ( aa_set_nat_o(member_nat(Y3),A4)
             => ( X = Y3 ) ) )
       => aa_set_nat_o(is_singleton_nat,A4) ) ) ).

% is_singletonI'
tff(fact_2213_is__singletonI_H,axiom,
    ! [A4: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int,Y3: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ( aa_set_int_o(member_int(Y3),A4)
             => ( X = Y3 ) ) )
       => aa_set_int_o(is_singleton_int,A4) ) ) ).

% is_singletonI'
tff(fact_2214_is__singletonI_H,axiom,
    ! [A4: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o,Y3: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ( aa_set_o_o(member_o((Y3)),A4)
             => ( (X)
              <=> (Y3) ) ) )
       => aa_set_o_o(is_singleton_o,A4) ) ) ).

% is_singletonI'
tff(fact_2215_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,T)),one_one_nat) ).

% minNull_bound
tff(fact_2216_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf($false,$false)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_2217_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv2: $o] : ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf($true,(Uv2))) = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_2218_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu2: $o] : ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf((Uu2),$true)) = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_2219_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,vEBT_Leaf((A),(B))),Xa) = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_2220_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,vEBT_Leaf((Uu2),(Uv2))),zero_zero_nat) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_2221_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Na: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,Na)) = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_2222_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_2223_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,vEBT_Leaf((Uu2),(Uv2))),zero_zero_nat) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_2224_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz),Va2)),Vb) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_2225_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Na: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,Na)) = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_2226_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz),Va2)),Vb) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_2227_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz)),Va2) = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_2228_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,vEBT_Leaf((Uu2),(B))),zero_zero_nat) = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_2229_is__singletonE,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(is_sin24926331636114728T_VEBT,A4)
     => ~ ! [X: vEBT_VEBT] : ( A4 != aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),bot_bo8194388402131092736T_VEBT) ) ) ).

% is_singletonE
tff(fact_2230_is__singletonE,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
     => ~ ! [X: nat] : ( A4 != aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ) ).

% is_singletonE
tff(fact_2231_is__singletonE,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_singleton_int,A4)
     => ~ ! [X: int] : ( A4 != aa_set_int_set_int(insert_int(X),bot_bot_set_int) ) ) ).

% is_singletonE
tff(fact_2232_is__singletonE,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_singleton_o,A4)
     => ~ ! [X: $o] : ( A4 != aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ) ) ).

% is_singletonE
tff(fact_2233_is__singleton__def,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(is_sin24926331636114728T_VEBT,A4)
    <=> ? [X5: vEBT_VEBT] : ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X5),bot_bo8194388402131092736T_VEBT) ) ) ).

% is_singleton_def
tff(fact_2234_is__singleton__def,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
    <=> ? [X5: nat] : ( A4 = aa_set_nat_set_nat(insert_nat(X5),bot_bot_set_nat) ) ) ).

% is_singleton_def
tff(fact_2235_is__singleton__def,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(is_singleton_int,A4)
    <=> ? [X5: int] : ( A4 = aa_set_int_set_int(insert_int(X5),bot_bot_set_int) ) ) ).

% is_singleton_def
tff(fact_2236_is__singleton__def,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(is_singleton_o,A4)
    <=> ? [X5: $o] : ( A4 = aa_set_o_set_o(insert_o((X5)),bot_bot_set_o) ) ) ).

% is_singleton_def
tff(fact_2237_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz)),Va2) = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_2238_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb),Vc)) = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_2239_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2),Uy2)) = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_2240_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xa) = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_2241_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)),Vf2) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_2242_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,vEBT_Leaf((A),(Uw2))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_2243_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2)),Ve2) = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_2244_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)),Vf2) = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_2245_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd2)),Ve2) = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_2246_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz)),Xa) = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_2247_set__removeAll,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,removeAll_VEBT_VEBT(Xa,Xs)) = minus_5127226145743854075T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)) ) ).

% set_removeAll
tff(fact_2248_set__removeAll,axiom,
    ! [Xa: real,Xs: list_real] : ( aa_lis3769085826154824284t_real(set_real2,removeAll_real(Xa,Xs)) = minus_minus_set_real(aa_lis3769085826154824284t_real(set_real2,Xs),aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)) ) ).

% set_removeAll
tff(fact_2249_set__removeAll,axiom,
    ! [Xa: int,Xs: list_int] : ( aa_list_int_set_int(set_int2,removeAll_int(Xa,Xs)) = minus_minus_set_int(aa_list_int_set_int(set_int2,Xs),aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) ) ).

% set_removeAll
tff(fact_2250_set__removeAll,axiom,
    ! [Xa: $o,Xs: list_o] : ( aa_list_o_set_o(set_o2,removeAll_o((Xa),Xs)) = minus_minus_set_o(aa_list_o_set_o(set_o2,Xs),aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)) ) ).

% set_removeAll
tff(fact_2251_set__removeAll,axiom,
    ! [Xa: nat,Xs: list_nat] : ( aa_list_nat_set_nat(set_nat2,removeAll_nat(Xa,Xs)) = minus_minus_set_nat(aa_list_nat_set_nat(set_nat2,Xs),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) ) ).

% set_removeAll
tff(fact_2252_vebt__maxt_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      some_nat(one_one_nat),
                      $ite((A3),some_nat(zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2) )
                 => ( ( Y = some_nat(Ma2) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_2253_vebt__mint_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      (A3),
                      some_nat(zero_zero_nat),
                      $ite((B3),some_nat(one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2) )
                 => ( ( Y = some_nat(Mi2) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_2254_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary))) ) ).

% height_node
tff(fact_2255_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,T),L)),one_one_nat) ) ) ).

% insertsimp'
tff(fact_2256_insersimp_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),X_1)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,T),Y)),one_one_nat) ) ) ).

% insersimp'
tff(fact_2257_frac__unique__iff,axiom,
    ! [Xa: real,A: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xa) = A )
    <=> ( aa_set_real_o(member_real(aa_real_real(minus_minus_real(Xa),A)),ring_1_Ints_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) ) ) ).

% frac_unique_iff
tff(fact_2258_frac__unique__iff,axiom,
    ! [Xa: rat,A: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xa) = A )
    <=> ( aa_set_rat_o(member_rat(aa_rat_rat(minus_minus_rat(Xa),A)),ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) ) ) ).

% frac_unique_iff
tff(fact_2259_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary)),Xa) = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_2260_remove__def,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT] : ( remove_VEBT_VEBT(Xa,A4) = minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)) ) ).

% remove_def
tff(fact_2261_remove__def,axiom,
    ! [Xa: int,A4: set_int] : ( remove_int(Xa,A4) = minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) ) ).

% remove_def
tff(fact_2262_remove__def,axiom,
    ! [Xa: $o,A4: set_o] : ( remove_o((Xa),A4) = minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)) ) ).

% remove_def
tff(fact_2263_remove__def,axiom,
    ! [Xa: nat,A4: set_nat] : ( remove_nat(Xa,A4) = minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) ) ).

% remove_def
tff(fact_2264_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,Sa: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),Sa)),Xa) = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_2265_member__remove,axiom,
    ! [Xa: nat,Y: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),remove_nat(Y,A4))
    <=> ( aa_set_nat_o(member_nat(Xa),A4)
        & ( Xa != Y ) ) ) ).

% member_remove
tff(fact_2266_member__remove,axiom,
    ! [Xa: vEBT_VEBT,Y: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),remove_VEBT_VEBT(Y,A4))
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
        & ( Xa != Y ) ) ) ).

% member_remove
tff(fact_2267_member__remove,axiom,
    ! [Xa: real,Y: real,A4: set_real] :
      ( aa_set_real_o(member_real(Xa),remove_real(Y,A4))
    <=> ( aa_set_real_o(member_real(Xa),A4)
        & ( Xa != Y ) ) ) ).

% member_remove
tff(fact_2268_member__remove,axiom,
    ! [Xa: int,Y: int,A4: set_int] :
      ( aa_set_int_o(member_int(Xa),remove_int(Y,A4))
    <=> ( aa_set_int_o(member_int(Xa),A4)
        & ( Xa != Y ) ) ) ).

% member_remove
tff(fact_2269_member__remove,axiom,
    ! [Xa: set_nat,Y: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),remove_set_nat(Y,A4))
    <=> ( aa_set_set_nat_o(member_set_nat(Xa),A4)
        & ( Xa != Y ) ) ) ).

% member_remove
tff(fact_2270_frac__eq__0__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xa) = zero_zero_real )
    <=> aa_set_real_o(member_real(Xa),ring_1_Ints_real) ) ).

% frac_eq_0_iff
tff(fact_2271_frac__eq__0__iff,axiom,
    ! [Xa: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xa) = zero_zero_rat )
    <=> aa_set_rat_o(member_rat(Xa),ring_1_Ints_rat) ) ).

% frac_eq_0_iff
tff(fact_2272_frac__gt__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xa))
    <=> ~ aa_set_real_o(member_real(Xa),ring_1_Ints_real) ) ).

% frac_gt_0_iff
tff(fact_2273_frac__gt__0__iff,axiom,
    ! [Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,Xa))
    <=> ~ aa_set_rat_o(member_rat(Xa),ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_2274_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Leaf((A),(B))) = zero_zero_nat ) ).

% VEBT_internal.height.simps(1)
tff(fact_2275_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,vEBT_Leaf((A),(B))),Xa) = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_2276_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,Sa: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),Sa)),Xa) = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_2277_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one_nat )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xa = vEBT_Leaf($true,(Uv)) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu: $o] :
                  ( ( Xa = vEBT_Leaf((Uu),$true) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf((Uu),$true)) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_2278_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% delete_bound_height'
tff(fact_2279_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary))) ) ).

% height_compose_child
tff(fact_2280_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Summary),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))))) ) ).

% height_compose_list
tff(fact_2281_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Summary))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary))) ).

% height_compose_summary
tff(fact_2282_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
      <=> (Y) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xa = vEBT_Leaf($true,(Uv)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu: $o] :
                  ( ( Xa = vEBT_Leaf((Uu),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_2283_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux),Uy)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_2284_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xa)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xa)
       => ( ! [Uv: $o] :
              ( ( Xa = vEBT_Leaf($true,(Uv)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) )
         => ( ! [Uu: $o] :
                ( ( Xa = vEBT_Leaf((Uu),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_2285_Ints__0,axiom,
    member_uint32(zero_zero_uint32,ring_1_Ints_uint32) ).

% Ints_0
tff(fact_2286_Ints__0,axiom,
    aa_set_real_o(member_real(zero_zero_real),ring_1_Ints_real) ).

% Ints_0
tff(fact_2287_Ints__0,axiom,
    aa_set_rat_o(member_rat(zero_zero_rat),ring_1_Ints_rat) ).

% Ints_0
tff(fact_2288_Ints__0,axiom,
    aa_set_int_o(member_int(zero_zero_int),ring_1_Ints_int) ).

% Ints_0
tff(fact_2289_enumerate__Suc_H,axiom,
    ! [S2: set_nat,Na: nat] : ( aa_nat_nat(infini8530281810654367211te_nat(S2),aa_nat_nat(suc,Na)) = aa_nat_nat(infini8530281810654367211te_nat(minus_minus_set_nat(S2,aa_set_nat_set_nat(insert_nat(aa_nat_nat(infini8530281810654367211te_nat(S2),zero_zero_nat)),bot_bot_set_nat))),Na) ) ).

% enumerate_Suc'
tff(fact_2290_even__odd__cases,axiom,
    ! [Xa: nat] :
      ( ! [N: nat] : ( Xa != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N) )
     => ~ ! [N: nat] : ( Xa != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ) ).

% even_odd_cases
tff(fact_2291_add__right__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_2292_add__right__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_2293_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_2294_add__right__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B = C2 ) ) ).

% add_right_cancel
tff(fact_2295_add__left__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_2296_add__left__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_2297_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_2298_add__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B = C2 ) ) ).

% add_left_cancel
tff(fact_2299_image__eqI,axiom,
    ! [B: nat,F: fun_nat_nat,Xa: nat,A4: set_nat] :
      ( ( B = aa_nat_nat(F,Xa) )
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ) ).

% image_eqI
tff(fact_2300_image__eqI,axiom,
    ! [B: vEBT_VEBT,F: fun_nat_VEBT_VEBT,Xa: nat,A4: set_nat] :
      ( ( B = aa_nat_VEBT_VEBT(F,Xa) )
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_nat_VEBT_VEBT(F,A4)) ) ) ).

% image_eqI
tff(fact_2301_image__eqI,axiom,
    ! [B: real,F: fun_nat_real,Xa: nat,A4: set_nat] :
      ( ( B = aa_nat_real(F,Xa) )
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_real_o(member_real(B),image_nat_real(F,A4)) ) ) ).

% image_eqI
tff(fact_2302_image__eqI,axiom,
    ! [B: int,F: fun_nat_int,Xa: nat,A4: set_nat] :
      ( ( B = aa_nat_int(F,Xa) )
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_int_o(member_int(B),aa_set_nat_set_int(image_nat_int(F),A4)) ) ) ).

% image_eqI
tff(fact_2303_image__eqI,axiom,
    ! [B: nat,F: fun_VEBT_VEBT_nat,Xa: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( B = aa_VEBT_VEBT_nat(F,Xa) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => aa_set_nat_o(member_nat(B),image_VEBT_VEBT_nat(F,A4)) ) ) ).

% image_eqI
tff(fact_2304_image__eqI,axiom,
    ! [B: vEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,Xa: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( B = aa_VEB5832953974749077948T_VEBT(F,Xa) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_3375948659692109573T_VEBT(F,A4)) ) ) ).

% image_eqI
tff(fact_2305_image__eqI,axiom,
    ! [B: real,F: fun_VEBT_VEBT_real,Xa: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( B = aa_VEBT_VEBT_real(F,Xa) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => aa_set_real_o(member_real(B),image_VEBT_VEBT_real(F,A4)) ) ) ).

% image_eqI
tff(fact_2306_image__eqI,axiom,
    ! [B: int,F: fun_VEBT_VEBT_int,Xa: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ( B = aa_VEBT_VEBT_int(F,Xa) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => aa_set_int_o(member_int(B),image_VEBT_VEBT_int(F,A4)) ) ) ).

% image_eqI
tff(fact_2307_image__eqI,axiom,
    ! [B: nat,F: fun_real_nat,Xa: real,A4: set_real] :
      ( ( B = aa_real_nat(F,Xa) )
     => ( aa_set_real_o(member_real(Xa),A4)
       => aa_set_nat_o(member_nat(B),image_real_nat(F,A4)) ) ) ).

% image_eqI
tff(fact_2308_image__eqI,axiom,
    ! [B: vEBT_VEBT,F: fun_real_VEBT_VEBT,Xa: real,A4: set_real] :
      ( ( B = aa_real_VEBT_VEBT(F,Xa) )
     => ( aa_set_real_o(member_real(Xa),A4)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_real_VEBT_VEBT(F,A4)) ) ) ).

% image_eqI
tff(fact_2309_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_2310_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_2311_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_2312_add_Oright__neutral,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(plus_plus_uint32(A),zero_zero_uint32) = A ) ).

% add.right_neutral
tff(fact_2313_add_Oright__neutral,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ) ).

% add.right_neutral
tff(fact_2314_add_Oright__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ) ).

% add.right_neutral
tff(fact_2315_add_Oright__neutral,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ) ).

% add.right_neutral
tff(fact_2316_add_Oright__neutral,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ) ).

% add.right_neutral
tff(fact_2317_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_2318_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_2319_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_2320_add__cancel__left__left,axiom,
    ! [B: uint32,A: uint32] :
      ( ( aa_uint32_uint32(plus_plus_uint32(B),A) = A )
    <=> ( B = zero_zero_uint32 ) ) ).

% add_cancel_left_left
tff(fact_2321_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_2322_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_2323_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_2324_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_2325_add__cancel__left__right,axiom,
    ! [A: uint32,B: uint32] :
      ( ( aa_uint32_uint32(plus_plus_uint32(A),B) = A )
    <=> ( B = zero_zero_uint32 ) ) ).

% add_cancel_left_right
tff(fact_2326_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_2327_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_2328_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_2329_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_2330_add__cancel__right__left,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A = aa_uint32_uint32(plus_plus_uint32(B),A) )
    <=> ( B = zero_zero_uint32 ) ) ).

% add_cancel_right_left
tff(fact_2331_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_2332_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_2333_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_2334_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_2335_add__cancel__right__right,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A = aa_uint32_uint32(plus_plus_uint32(A),B) )
    <=> ( B = zero_zero_uint32 ) ) ).

% add_cancel_right_right
tff(fact_2336_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_2337_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_2338_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_2339_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_2340_add__eq__0__iff__both__eq__0,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = zero_zero_nat )
    <=> ( ( Xa = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2341_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) )
    <=> ( ( Xa = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2342_add__0,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(plus_plus_uint32(zero_zero_uint32),A) = A ) ).

% add_0
tff(fact_2343_add__0,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ) ).

% add_0
tff(fact_2344_add__0,axiom,
    ! [A: rat] : ( aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ) ).

% add_0
tff(fact_2345_add__0,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ) ).

% add_0
tff(fact_2346_add__0,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ) ).

% add_0
tff(fact_2347_add__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_2348_add__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_2349_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_2350_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_2351_add__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_2352_add__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_2353_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_2354_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_2355_add__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_2356_add__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_2357_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_2358_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_2359_add__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_2360_add__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_2361_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_2362_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_2363_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ) ).

% add_diff_cancel_right'
tff(fact_2364_add__diff__cancel__right_H,axiom,
    ! [A: uint32,B: uint32] : ( minus_minus_uint32(aa_uint32_uint32(plus_plus_uint32(A),B),B) = A ) ).

% add_diff_cancel_right'
tff(fact_2365_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : ( aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ) ).

% add_diff_cancel_right'
tff(fact_2366_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ) ).

% add_diff_cancel_right'
tff(fact_2367_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ) ).

% add_diff_cancel_right'
tff(fact_2368_add__diff__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] : ( aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(A),B) ) ).

% add_diff_cancel_right
tff(fact_2369_add__diff__cancel__right,axiom,
    ! [A: uint32,C2: uint32,B: uint32] : ( minus_minus_uint32(aa_uint32_uint32(plus_plus_uint32(A),C2),aa_uint32_uint32(plus_plus_uint32(B),C2)) = minus_minus_uint32(A,B) ) ).

% add_diff_cancel_right
tff(fact_2370_add__diff__cancel__right,axiom,
    ! [A: real,C2: real,B: real] : ( aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(minus_minus_real(A),B) ) ).

% add_diff_cancel_right
tff(fact_2371_add__diff__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ).

% add_diff_cancel_right
tff(fact_2372_add__diff__cancel__right,axiom,
    ! [A: int,C2: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ).

% add_diff_cancel_right
tff(fact_2373_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ) ).

% add_diff_cancel_left'
tff(fact_2374_add__diff__cancel__left_H,axiom,
    ! [A: uint32,B: uint32] : ( minus_minus_uint32(aa_uint32_uint32(plus_plus_uint32(A),B),A) = B ) ).

% add_diff_cancel_left'
tff(fact_2375_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : ( aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ) ).

% add_diff_cancel_left'
tff(fact_2376_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ) ).

% add_diff_cancel_left'
tff(fact_2377_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ) ).

% add_diff_cancel_left'
tff(fact_2378_add__diff__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] : ( aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)) = aa_rat_rat(minus_minus_rat(A),B) ) ).

% add_diff_cancel_left
tff(fact_2379_add__diff__cancel__left,axiom,
    ! [C2: uint32,A: uint32,B: uint32] : ( minus_minus_uint32(aa_uint32_uint32(plus_plus_uint32(C2),A),aa_uint32_uint32(plus_plus_uint32(C2),B)) = minus_minus_uint32(A,B) ) ).

% add_diff_cancel_left
tff(fact_2380_add__diff__cancel__left,axiom,
    ! [C2: real,A: real,B: real] : ( aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) = aa_real_real(minus_minus_real(A),B) ) ).

% add_diff_cancel_left
tff(fact_2381_add__diff__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ).

% add_diff_cancel_left
tff(fact_2382_add__diff__cancel__left,axiom,
    ! [C2: int,A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ).

% add_diff_cancel_left
tff(fact_2383_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ).

% diff_add_cancel
tff(fact_2384_diff__add__cancel,axiom,
    ! [A: uint32,B: uint32] : ( aa_uint32_uint32(plus_plus_uint32(minus_minus_uint32(A,B)),B) = A ) ).

% diff_add_cancel
tff(fact_2385_diff__add__cancel,axiom,
    ! [A: real,B: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A),B)),B) = A ) ).

% diff_add_cancel
tff(fact_2386_diff__add__cancel,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ).

% diff_add_cancel
tff(fact_2387_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ) ).

% add_diff_cancel
tff(fact_2388_add__diff__cancel,axiom,
    ! [A: uint32,B: uint32] : ( minus_minus_uint32(aa_uint32_uint32(plus_plus_uint32(A),B),B) = A ) ).

% add_diff_cancel
tff(fact_2389_add__diff__cancel,axiom,
    ! [A: real,B: real] : ( aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ) ).

% add_diff_cancel
tff(fact_2390_add__diff__cancel,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ) ).

% add_diff_cancel
tff(fact_2391_image__empty,axiom,
    ! [F: fun_nat_set_nat] : ( aa_set7246812951530217472et_nat(image_nat_set_nat(F),bot_bot_set_nat) = bot_bot_set_set_nat ) ).

% image_empty
tff(fact_2392_image__empty,axiom,
    ! [F: fun_nat_nat] : ( aa_set_nat_set_nat(image_nat_nat(F),bot_bot_set_nat) = bot_bot_set_nat ) ).

% image_empty
tff(fact_2393_image__empty,axiom,
    ! [F: fun_nat_int] : ( aa_set_nat_set_int(image_nat_int(F),bot_bot_set_nat) = bot_bot_set_int ) ).

% image_empty
tff(fact_2394_image__empty,axiom,
    ! [F: fun_nat_o] : ( image_nat_o(F,bot_bot_set_nat) = bot_bot_set_o ) ).

% image_empty
tff(fact_2395_image__empty,axiom,
    ! [F: fun_int_nat] : ( aa_set_int_set_nat(image_int_nat(F),bot_bot_set_int) = bot_bot_set_nat ) ).

% image_empty
tff(fact_2396_image__empty,axiom,
    ! [F: fun_int_int] : ( aa_set_int_set_int(image_int_int(F),bot_bot_set_int) = bot_bot_set_int ) ).

% image_empty
tff(fact_2397_image__empty,axiom,
    ! [F: fun_int_o] : ( image_int_o(F,bot_bot_set_int) = bot_bot_set_o ) ).

% image_empty
tff(fact_2398_image__empty,axiom,
    ! [F: fun_o_nat] : ( image_o_nat(F,bot_bot_set_o) = bot_bot_set_nat ) ).

% image_empty
tff(fact_2399_image__empty,axiom,
    ! [F: fun_o_int] : ( image_o_int(F,bot_bot_set_o) = bot_bot_set_int ) ).

% image_empty
tff(fact_2400_image__empty,axiom,
    ! [F: fun_o_o] : ( image_o_o(F,bot_bot_set_o) = bot_bot_set_o ) ).

% image_empty
tff(fact_2401_empty__is__image,axiom,
    ! [F: fun_nat_set_nat,A4: set_nat] :
      ( ( bot_bot_set_set_nat = aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4) )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_2402_empty__is__image,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(image_nat_nat(F),A4) )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_2403_empty__is__image,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( ( bot_bot_set_nat = aa_set_int_set_nat(image_int_nat(F),A4) )
    <=> ( A4 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_2404_empty__is__image,axiom,
    ! [F: fun_o_nat,A4: set_o] :
      ( ( bot_bot_set_nat = image_o_nat(F,A4) )
    <=> ( A4 = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_2405_empty__is__image,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( ( bot_bot_set_int = aa_set_nat_set_int(image_nat_int(F),A4) )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_2406_empty__is__image,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(image_int_int(F),A4) )
    <=> ( A4 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_2407_empty__is__image,axiom,
    ! [F: fun_o_int,A4: set_o] :
      ( ( bot_bot_set_int = image_o_int(F,A4) )
    <=> ( A4 = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_2408_empty__is__image,axiom,
    ! [F: fun_nat_o,A4: set_nat] :
      ( ( bot_bot_set_o = image_nat_o(F,A4) )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_2409_empty__is__image,axiom,
    ! [F: fun_int_o,A4: set_int] :
      ( ( bot_bot_set_o = image_int_o(F,A4) )
    <=> ( A4 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_2410_empty__is__image,axiom,
    ! [F: fun_o_o,A4: set_o] :
      ( ( bot_bot_set_o = image_o_o(F,A4) )
    <=> ( A4 = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_2411_image__is__empty,axiom,
    ! [F: fun_nat_set_nat,A4: set_nat] :
      ( ( aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4) = bot_bot_set_set_nat )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_2412_image__is__empty,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( ( aa_set_nat_set_nat(image_nat_nat(F),A4) = bot_bot_set_nat )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_2413_image__is__empty,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( ( aa_set_int_set_nat(image_int_nat(F),A4) = bot_bot_set_nat )
    <=> ( A4 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_2414_image__is__empty,axiom,
    ! [F: fun_o_nat,A4: set_o] :
      ( ( image_o_nat(F,A4) = bot_bot_set_nat )
    <=> ( A4 = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_2415_image__is__empty,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( ( aa_set_nat_set_int(image_nat_int(F),A4) = bot_bot_set_int )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_2416_image__is__empty,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( ( aa_set_int_set_int(image_int_int(F),A4) = bot_bot_set_int )
    <=> ( A4 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_2417_image__is__empty,axiom,
    ! [F: fun_o_int,A4: set_o] :
      ( ( image_o_int(F,A4) = bot_bot_set_int )
    <=> ( A4 = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_2418_image__is__empty,axiom,
    ! [F: fun_nat_o,A4: set_nat] :
      ( ( image_nat_o(F,A4) = bot_bot_set_o )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_2419_image__is__empty,axiom,
    ! [F: fun_int_o,A4: set_int] :
      ( ( image_int_o(F,A4) = bot_bot_set_o )
    <=> ( A4 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_2420_image__is__empty,axiom,
    ! [F: fun_o_o,A4: set_o] :
      ( ( image_o_o(F,A4) = bot_bot_set_o )
    <=> ( A4 = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_2421_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ) ).

% add_Suc_right
tff(fact_2422_finite__imageI,axiom,
    ! [F2: set_nat,H2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(H2),F2)) ) ).

% finite_imageI
tff(fact_2423_finite__imageI,axiom,
    ! [F2: set_nat,H2: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(H2),F2)) ) ).

% finite_imageI
tff(fact_2424_finite__imageI,axiom,
    ! [F2: set_nat,H2: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => aa_set_complex_o(finite3207457112153483333omplex,image_nat_complex(H2,F2)) ) ).

% finite_imageI
tff(fact_2425_finite__imageI,axiom,
    ! [F2: set_nat,H2: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_1215581382706833972nteger(H2,F2)) ) ).

% finite_imageI
tff(fact_2426_finite__imageI,axiom,
    ! [F2: set_int,H2: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,F2)
     => aa_set_nat_o(finite_finite_nat,aa_set_int_set_nat(image_int_nat(H2),F2)) ) ).

% finite_imageI
tff(fact_2427_finite__imageI,axiom,
    ! [F2: set_int,H2: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,F2)
     => aa_set_int_o(finite_finite_int,aa_set_int_set_int(image_int_int(H2),F2)) ) ).

% finite_imageI
tff(fact_2428_finite__imageI,axiom,
    ! [F2: set_int,H2: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,F2)
     => aa_set_complex_o(finite3207457112153483333omplex,image_int_complex(H2,F2)) ) ).

% finite_imageI
tff(fact_2429_finite__imageI,axiom,
    ! [F2: set_int,H2: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,F2)
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_1587234942943678608nteger(H2,F2)) ) ).

% finite_imageI
tff(fact_2430_finite__imageI,axiom,
    ! [F2: set_complex,H2: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => aa_set_nat_o(finite_finite_nat,image_complex_nat(H2,F2)) ) ).

% finite_imageI
tff(fact_2431_finite__imageI,axiom,
    ! [F2: set_complex,H2: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => aa_set_int_o(finite_finite_int,image_complex_int(H2,F2)) ) ).

% finite_imageI
tff(fact_2432_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( Na = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_2433_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_2434_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_2435_insert__image,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_set_nat(insert_nat(aa_nat_nat(F,Xa)),aa_set_nat_set_nat(image_nat_nat(F),A4)) = aa_set_nat_set_nat(image_nat_nat(F),A4) ) ) ).

% insert_image
tff(fact_2436_insert__image,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_nat_VEBT_VEBT(F,Xa)),image_nat_VEBT_VEBT(F,A4)) = image_nat_VEBT_VEBT(F,A4) ) ) ).

% insert_image
tff(fact_2437_insert__image,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_int_set_int(insert_int(aa_nat_int(F,Xa)),aa_set_nat_set_int(image_nat_int(F),A4)) = aa_set_nat_set_int(image_nat_int(F),A4) ) ) ).

% insert_image
tff(fact_2438_insert__image,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_o_set_o(insert_o(aa_nat_o(F,Xa)),image_nat_o(F,A4)) = image_nat_o(F,A4) ) ) ).

% insert_image
tff(fact_2439_insert__image,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( aa_set_nat_set_nat(insert_nat(aa_VEBT_VEBT_nat(F,Xa)),image_VEBT_VEBT_nat(F,A4)) = image_VEBT_VEBT_nat(F,A4) ) ) ).

% insert_image
tff(fact_2440_insert__image,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(F,Xa)),image_3375948659692109573T_VEBT(F,A4)) = image_3375948659692109573T_VEBT(F,A4) ) ) ).

% insert_image
tff(fact_2441_insert__image,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( aa_set_int_set_int(insert_int(aa_VEBT_VEBT_int(F,Xa)),image_VEBT_VEBT_int(F,A4)) = image_VEBT_VEBT_int(F,A4) ) ) ).

% insert_image
tff(fact_2442_insert__image,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( aa_set_o_set_o(insert_o(aa_VEBT_VEBT_o(F,Xa)),image_VEBT_VEBT_o(F,A4)) = image_VEBT_VEBT_o(F,A4) ) ) ).

% insert_image
tff(fact_2443_insert__image,axiom,
    ! [Xa: real,A4: set_real,F: fun_real_nat] :
      ( aa_set_real_o(member_real(Xa),A4)
     => ( aa_set_nat_set_nat(insert_nat(aa_real_nat(F,Xa)),image_real_nat(F,A4)) = image_real_nat(F,A4) ) ) ).

% insert_image
tff(fact_2444_insert__image,axiom,
    ! [Xa: real,A4: set_real,F: fun_real_VEBT_VEBT] :
      ( aa_set_real_o(member_real(Xa),A4)
     => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_real_VEBT_VEBT(F,Xa)),image_real_VEBT_VEBT(F,A4)) = image_real_VEBT_VEBT(F,A4) ) ) ).

% insert_image
tff(fact_2445_image__insert,axiom,
    ! [F: fun_nat_nat,A: nat,B4: set_nat] : ( aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(insert_nat(A),B4)) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(F,A)),aa_set_nat_set_nat(image_nat_nat(F),B4)) ) ).

% image_insert
tff(fact_2446_image__insert,axiom,
    ! [F: fun_nat_VEBT_VEBT,A: nat,B4: set_nat] : ( image_nat_VEBT_VEBT(F,aa_set_nat_set_nat(insert_nat(A),B4)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_nat_VEBT_VEBT(F,A)),image_nat_VEBT_VEBT(F,B4)) ) ).

% image_insert
tff(fact_2447_image__insert,axiom,
    ! [F: fun_nat_int,A: nat,B4: set_nat] : ( aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(insert_nat(A),B4)) = aa_set_int_set_int(insert_int(aa_nat_int(F,A)),aa_set_nat_set_int(image_nat_int(F),B4)) ) ).

% image_insert
tff(fact_2448_image__insert,axiom,
    ! [F: fun_nat_o,A: nat,B4: set_nat] : ( image_nat_o(F,aa_set_nat_set_nat(insert_nat(A),B4)) = aa_set_o_set_o(insert_o(aa_nat_o(F,A)),image_nat_o(F,B4)) ) ).

% image_insert
tff(fact_2449_image__insert,axiom,
    ! [F: fun_VEBT_VEBT_nat,A: vEBT_VEBT,B4: set_VEBT_VEBT] : ( image_VEBT_VEBT_nat(F,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4)) = aa_set_nat_set_nat(insert_nat(aa_VEBT_VEBT_nat(F,A)),image_VEBT_VEBT_nat(F,B4)) ) ).

% image_insert
tff(fact_2450_image__insert,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,A: vEBT_VEBT,B4: set_VEBT_VEBT] : ( image_3375948659692109573T_VEBT(F,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(F,A)),image_3375948659692109573T_VEBT(F,B4)) ) ).

% image_insert
tff(fact_2451_image__insert,axiom,
    ! [F: fun_VEBT_VEBT_int,A: vEBT_VEBT,B4: set_VEBT_VEBT] : ( image_VEBT_VEBT_int(F,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4)) = aa_set_int_set_int(insert_int(aa_VEBT_VEBT_int(F,A)),image_VEBT_VEBT_int(F,B4)) ) ).

% image_insert
tff(fact_2452_image__insert,axiom,
    ! [F: fun_VEBT_VEBT_o,A: vEBT_VEBT,B4: set_VEBT_VEBT] : ( image_VEBT_VEBT_o(F,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4)) = aa_set_o_set_o(insert_o(aa_VEBT_VEBT_o(F,A)),image_VEBT_VEBT_o(F,B4)) ) ).

% image_insert
tff(fact_2453_image__insert,axiom,
    ! [F: fun_int_nat,A: int,B4: set_int] : ( aa_set_int_set_nat(image_int_nat(F),aa_set_int_set_int(insert_int(A),B4)) = aa_set_nat_set_nat(insert_nat(aa_int_nat(F,A)),aa_set_int_set_nat(image_int_nat(F),B4)) ) ).

% image_insert
tff(fact_2454_image__insert,axiom,
    ! [F: fun_int_VEBT_VEBT,A: int,B4: set_int] : ( image_int_VEBT_VEBT(F,aa_set_int_set_int(insert_int(A),B4)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_int_VEBT_VEBT(F,A)),image_int_VEBT_VEBT(F,B4)) ) ).

% image_insert
tff(fact_2455_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_2456_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_2457_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2458_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2459_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2460_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2461_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2462_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2463_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_2464_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_2465_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_2466_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_2467_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_2468_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_2469_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_2470_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_2471_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_2472_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_2473_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_2474_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_2475_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_2476_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_2477_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_2478_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_2479_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2480_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2481_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2482_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2483_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2484_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2485_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_2486_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_2487_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_2488_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_2489_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_2490_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_2491_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_2492_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_2493_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_2494_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_2495_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_2496_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_2497_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_2498_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_2499_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_2500_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_2501_diff__add__zero,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ) ).

% diff_add_zero
tff(fact_2502_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2503_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2504_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2505_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2506_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2507_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2508_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2509_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2510_image__add__0,axiom,
    ! [S2: set_uint32] : ( image_uint32_uint32(plus_plus_uint32(zero_zero_uint32),S2) = S2 ) ).

% image_add_0
tff(fact_2511_image__add__0,axiom,
    ! [S2: set_real] : ( image_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),S2) = S2 ) ).

% image_add_0
tff(fact_2512_image__add__0,axiom,
    ! [S2: set_rat] : ( image_rat_rat(plus_plus_rat(zero_zero_rat),S2) = S2 ) ).

% image_add_0
tff(fact_2513_image__add__0,axiom,
    ! [S2: set_nat] : ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat)),S2) = S2 ) ).

% image_add_0
tff(fact_2514_image__add__0,axiom,
    ! [S2: set_int] : ( aa_set_int_set_int(image_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int)),S2) = S2 ) ).

% image_add_0
tff(fact_2515_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% add_gr_0
tff(fact_2516_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_2517_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_2518_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_2519_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_2520_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_2521_enumerate__mono__iff,axiom,
    ! [S2: set_nat,M: nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S2),M)),aa_nat_nat(infini8530281810654367211te_nat(S2),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% enumerate_mono_iff
tff(fact_2522_add__right__imp__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_2523_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_2524_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_2525_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
tff(fact_2526_add__left__imp__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_2527_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_2528_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_2529_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
tff(fact_2530_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2531_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] : ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C2)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2532_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2533_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2534_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: real,B: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_2535_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_2536_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_2537_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_2538_add_Oright__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_2539_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_2540_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A) )
    <=> ( B = C2 ) ) ).

% add.right_cancel
tff(fact_2541_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_2542_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_2543_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2) )
    <=> ( B = C2 ) ) ).

% add.left_cancel
tff(fact_2544_add_Oassoc,axiom,
    ! [A: real,B: real,C2: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ) ).

% add.assoc
tff(fact_2545_add_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] : ( aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ) ).

% add.assoc
tff(fact_2546_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% add.assoc
tff(fact_2547_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% add.assoc
tff(fact_2548_group__cancel_Oadd2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2549_group__cancel_Oadd2,axiom,
    ! [B4: rat,K: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B4) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_2550_group__cancel_Oadd2,axiom,
    ! [B4: nat,K: nat,B: nat,A: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2551_group__cancel_Oadd2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2552_group__cancel_Oadd1,axiom,
    ! [A4: real,K: real,A: real,B: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A4),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2553_group__cancel_Oadd1,axiom,
    ! [A4: rat,K: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(plus_plus_rat(A4),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_2554_group__cancel_Oadd1,axiom,
    ! [A4: nat,K: nat,A: nat,B: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2555_group__cancel_Oadd1,axiom,
    ! [A4: int,K: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2556_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_2557_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2558_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_2559_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_2560_rev__image__eqI,axiom,
    ! [Xa: nat,A4: set_nat,B: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( ( B = aa_nat_nat(F,Xa) )
       => aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ) ).

% rev_image_eqI
tff(fact_2561_rev__image__eqI,axiom,
    ! [Xa: nat,A4: set_nat,B: vEBT_VEBT,F: fun_nat_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( ( B = aa_nat_VEBT_VEBT(F,Xa) )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_nat_VEBT_VEBT(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_2562_rev__image__eqI,axiom,
    ! [Xa: nat,A4: set_nat,B: real,F: fun_nat_real] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( ( B = aa_nat_real(F,Xa) )
       => aa_set_real_o(member_real(B),image_nat_real(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_2563_rev__image__eqI,axiom,
    ! [Xa: nat,A4: set_nat,B: int,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( ( B = aa_nat_int(F,Xa) )
       => aa_set_int_o(member_int(B),aa_set_nat_set_int(image_nat_int(F),A4)) ) ) ).

% rev_image_eqI
tff(fact_2564_rev__image__eqI,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,B: nat,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( ( B = aa_VEBT_VEBT_nat(F,Xa) )
       => aa_set_nat_o(member_nat(B),image_VEBT_VEBT_nat(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_2565_rev__image__eqI,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( ( B = aa_VEB5832953974749077948T_VEBT(F,Xa) )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_3375948659692109573T_VEBT(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_2566_rev__image__eqI,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,B: real,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( ( B = aa_VEBT_VEBT_real(F,Xa) )
       => aa_set_real_o(member_real(B),image_VEBT_VEBT_real(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_2567_rev__image__eqI,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,B: int,F: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( ( B = aa_VEBT_VEBT_int(F,Xa) )
       => aa_set_int_o(member_int(B),image_VEBT_VEBT_int(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_2568_rev__image__eqI,axiom,
    ! [Xa: real,A4: set_real,B: nat,F: fun_real_nat] :
      ( aa_set_real_o(member_real(Xa),A4)
     => ( ( B = aa_real_nat(F,Xa) )
       => aa_set_nat_o(member_nat(B),image_real_nat(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_2569_rev__image__eqI,axiom,
    ! [Xa: real,A4: set_real,B: vEBT_VEBT,F: fun_real_VEBT_VEBT] :
      ( aa_set_real_o(member_real(Xa),A4)
     => ( ( B = aa_real_VEBT_VEBT(F,Xa) )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_real_VEBT_VEBT(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_2570_ball__imageD,axiom,
    ! [F: fun_nat_set_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4))
         => aa_set_nat_o(P2,X) )
     => ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_set_nat_o(P2,aa_nat_set_nat(F,X3)) ) ) ).

% ball_imageD
tff(fact_2571_ball__imageD,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(image_nat_nat(F),A4))
         => aa_nat_o(P2,X) )
     => ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_nat_o(P2,aa_nat_nat(F,X3)) ) ) ).

% ball_imageD
tff(fact_2572_ball__imageD,axiom,
    ! [F: fun_nat_int,A4: set_nat,P2: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),aa_set_nat_set_int(image_nat_int(F),A4))
         => aa_int_o(P2,X) )
     => ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A4)
         => aa_int_o(P2,aa_nat_int(F,X3)) ) ) ).

% ball_imageD
tff(fact_2573_ball__imageD,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_set_int_set_nat(image_int_nat(F),A4))
         => aa_nat_o(P2,X) )
     => ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_nat_o(P2,aa_int_nat(F,X3)) ) ) ).

% ball_imageD
tff(fact_2574_ball__imageD,axiom,
    ! [F: fun_int_int,A4: set_int,P2: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),aa_set_int_set_int(image_int_int(F),A4))
         => aa_int_o(P2,X) )
     => ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A4)
         => aa_int_o(P2,aa_int_int(F,X3)) ) ) ).

% ball_imageD
tff(fact_2575_image__cong,axiom,
    ! [M7: set_nat,N6: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( ( M7 = N6 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),N6)
           => ( aa_nat_set_nat(F,X) = aa_nat_set_nat(G,X) ) )
       => ( aa_set7246812951530217472et_nat(image_nat_set_nat(F),M7) = aa_set7246812951530217472et_nat(image_nat_set_nat(G),N6) ) ) ) ).

% image_cong
tff(fact_2576_image__cong,axiom,
    ! [M7: set_nat,N6: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( M7 = N6 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),N6)
           => ( aa_nat_nat(F,X) = aa_nat_nat(G,X) ) )
       => ( aa_set_nat_set_nat(image_nat_nat(F),M7) = aa_set_nat_set_nat(image_nat_nat(G),N6) ) ) ) ).

% image_cong
tff(fact_2577_image__cong,axiom,
    ! [M7: set_nat,N6: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ( M7 = N6 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),N6)
           => ( aa_nat_int(F,X) = aa_nat_int(G,X) ) )
       => ( aa_set_nat_set_int(image_nat_int(F),M7) = aa_set_nat_set_int(image_nat_int(G),N6) ) ) ) ).

% image_cong
tff(fact_2578_image__cong,axiom,
    ! [M7: set_int,N6: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ( M7 = N6 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),N6)
           => ( aa_int_nat(F,X) = aa_int_nat(G,X) ) )
       => ( aa_set_int_set_nat(image_int_nat(F),M7) = aa_set_int_set_nat(image_int_nat(G),N6) ) ) ) ).

% image_cong
tff(fact_2579_image__cong,axiom,
    ! [M7: set_int,N6: set_int,F: fun_int_int,G: fun_int_int] :
      ( ( M7 = N6 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),N6)
           => ( aa_int_int(F,X) = aa_int_int(G,X) ) )
       => ( aa_set_int_set_int(image_int_int(F),M7) = aa_set_int_set_int(image_int_int(G),N6) ) ) ) ).

% image_cong
tff(fact_2580_bex__imageD,axiom,
    ! [F: fun_nat_set_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( ? [X3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X3),aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4))
          & aa_set_nat_o(P2,X3) )
     => ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
          & aa_set_nat_o(P2,aa_nat_set_nat(F,X)) ) ) ).

% bex_imageD
tff(fact_2581_bex__imageD,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_nat_o] :
      ( ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),aa_set_nat_set_nat(image_nat_nat(F),A4))
          & aa_nat_o(P2,X3) )
     => ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
          & aa_nat_o(P2,aa_nat_nat(F,X)) ) ) ).

% bex_imageD
tff(fact_2582_bex__imageD,axiom,
    ! [F: fun_nat_int,A4: set_nat,P2: fun_int_o] :
      ( ? [X3: int] :
          ( aa_set_int_o(member_int(X3),aa_set_nat_set_int(image_nat_int(F),A4))
          & aa_int_o(P2,X3) )
     => ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
          & aa_int_o(P2,aa_nat_int(F,X)) ) ) ).

% bex_imageD
tff(fact_2583_bex__imageD,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_nat_o] :
      ( ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),aa_set_int_set_nat(image_int_nat(F),A4))
          & aa_nat_o(P2,X3) )
     => ? [X: int] :
          ( aa_set_int_o(member_int(X),A4)
          & aa_nat_o(P2,aa_int_nat(F,X)) ) ) ).

% bex_imageD
tff(fact_2584_bex__imageD,axiom,
    ! [F: fun_int_int,A4: set_int,P2: fun_int_o] :
      ( ? [X3: int] :
          ( aa_set_int_o(member_int(X3),aa_set_int_set_int(image_int_int(F),A4))
          & aa_int_o(P2,X3) )
     => ? [X: int] :
          ( aa_set_int_o(member_int(X),A4)
          & aa_int_o(P2,aa_int_int(F,X)) ) ) ).

% bex_imageD
tff(fact_2585_image__iff,axiom,
    ! [Z: nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(Z),aa_set_nat_set_nat(image_nat_nat(F),A4))
    <=> ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
          & ( Z = aa_nat_nat(F,X5) ) ) ) ).

% image_iff
tff(fact_2586_image__iff,axiom,
    ! [Z: nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(member_nat(Z),aa_set_int_set_nat(image_int_nat(F),A4))
    <=> ? [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
          & ( Z = aa_int_nat(F,X5) ) ) ) ).

% image_iff
tff(fact_2587_image__iff,axiom,
    ! [Z: int,F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(member_int(Z),aa_set_nat_set_int(image_nat_int(F),A4))
    <=> ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
          & ( Z = aa_nat_int(F,X5) ) ) ) ).

% image_iff
tff(fact_2588_image__iff,axiom,
    ! [Z: int,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(member_int(Z),aa_set_int_set_int(image_int_int(F),A4))
    <=> ? [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
          & ( Z = aa_int_int(F,X5) ) ) ) ).

% image_iff
tff(fact_2589_image__iff,axiom,
    ! [Z: set_nat,F: fun_nat_set_nat,A4: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Z),aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4))
    <=> ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
          & ( Z = aa_nat_set_nat(F,X5) ) ) ) ).

% image_iff
tff(fact_2590_imageI,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => aa_set_nat_o(member_nat(aa_nat_nat(F,Xa)),aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ).

% imageI
tff(fact_2591_imageI,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_nat_VEBT_VEBT(F,Xa)),image_nat_VEBT_VEBT(F,A4)) ) ).

% imageI
tff(fact_2592_imageI,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => aa_set_real_o(member_real(aa_nat_real(F,Xa)),image_nat_real(F,A4)) ) ).

% imageI
tff(fact_2593_imageI,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => aa_set_int_o(member_int(aa_nat_int(F,Xa)),aa_set_nat_set_int(image_nat_int(F),A4)) ) ).

% imageI
tff(fact_2594_imageI,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => aa_set_nat_o(member_nat(aa_VEBT_VEBT_nat(F,Xa)),image_VEBT_VEBT_nat(F,A4)) ) ).

% imageI
tff(fact_2595_imageI,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(F,Xa)),image_3375948659692109573T_VEBT(F,A4)) ) ).

% imageI
tff(fact_2596_imageI,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => aa_set_real_o(member_real(aa_VEBT_VEBT_real(F,Xa)),image_VEBT_VEBT_real(F,A4)) ) ).

% imageI
tff(fact_2597_imageI,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => aa_set_int_o(member_int(aa_VEBT_VEBT_int(F,Xa)),image_VEBT_VEBT_int(F,A4)) ) ).

% imageI
tff(fact_2598_imageI,axiom,
    ! [Xa: real,A4: set_real,F: fun_real_nat] :
      ( aa_set_real_o(member_real(Xa),A4)
     => aa_set_nat_o(member_nat(aa_real_nat(F,Xa)),image_real_nat(F,A4)) ) ).

% imageI
tff(fact_2599_imageI,axiom,
    ! [Xa: real,A4: set_real,F: fun_real_VEBT_VEBT] :
      ( aa_set_real_o(member_real(Xa),A4)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(F,Xa)),image_real_VEBT_VEBT(F,A4)) ) ).

% imageI
tff(fact_2600_image__mono,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4)),aa_set7246812951530217472et_nat(image_nat_set_nat(F),B4)) ) ).

% image_mono
tff(fact_2601_image__mono,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B4)) ) ).

% image_mono
tff(fact_2602_image__mono,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
     => aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),aa_set_nat_set_int(image_nat_int(F),B4)) ) ).

% image_mono
tff(fact_2603_image__mono,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B4)) ) ).

% image_mono
tff(fact_2604_image__mono,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),aa_set_int_set_int(image_int_int(F),B4)) ) ).

% image_mono
tff(fact_2605_image__subsetI,axiom,
    ! [A4: set_nat,F: fun_nat_nat,B4: set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X)),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)),B4) ) ).

% image_subsetI
tff(fact_2606_image__subsetI,axiom,
    ! [A4: set_nat,F: fun_nat_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_nat_VEBT_VEBT(F,X)),B4) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(image_nat_VEBT_VEBT(F,A4)),B4) ) ).

% image_subsetI
tff(fact_2607_image__subsetI,axiom,
    ! [A4: set_nat,F: fun_nat_real,B4: set_real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_real_o(member_real(aa_nat_real(F,X)),B4) )
     => aa_set_real_o(ord_less_eq_set_real(image_nat_real(F,A4)),B4) ) ).

% image_subsetI
tff(fact_2608_image__subsetI,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,B4: set_nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_set_nat_o(member_nat(aa_VEBT_VEBT_nat(F,X)),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_VEBT_VEBT_nat(F,A4)),B4) ) ).

% image_subsetI
tff(fact_2609_image__subsetI,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(F,X)),B4) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(image_3375948659692109573T_VEBT(F,A4)),B4) ) ).

% image_subsetI
tff(fact_2610_image__subsetI,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,B4: set_real] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_set_real_o(member_real(aa_VEBT_VEBT_real(F,X)),B4) )
     => aa_set_real_o(ord_less_eq_set_real(image_VEBT_VEBT_real(F,A4)),B4) ) ).

% image_subsetI
tff(fact_2611_image__subsetI,axiom,
    ! [A4: set_real,F: fun_real_nat,B4: set_nat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_set_nat_o(member_nat(aa_real_nat(F,X)),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_real_nat(F,A4)),B4) ) ).

% image_subsetI
tff(fact_2612_image__subsetI,axiom,
    ! [A4: set_real,F: fun_real_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(F,X)),B4) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(image_real_VEBT_VEBT(F,A4)),B4) ) ).

% image_subsetI
tff(fact_2613_image__subsetI,axiom,
    ! [A4: set_real,F: fun_real_real,B4: set_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_set_real_o(member_real(aa_real_real(F,X)),B4) )
     => aa_set_real_o(ord_less_eq_set_real(image_real_real(F,A4)),B4) ) ).

% image_subsetI
tff(fact_2614_image__subsetI,axiom,
    ! [A4: set_int,F: fun_int_nat,B4: set_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_nat_o(member_nat(aa_int_nat(F,X)),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_int_set_nat(image_int_nat(F),A4)),B4) ) ).

% image_subsetI
tff(fact_2615_subset__imageE,axiom,
    ! [B4: set_set_nat,F: fun_nat_set_nat,A4: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B4),aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4))
     => ~ ! [C5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),A4)
           => ( B4 != aa_set7246812951530217472et_nat(image_nat_set_nat(F),C5) ) ) ) ).

% subset_imageE
tff(fact_2616_subset__imageE,axiom,
    ! [B4: set_nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ~ ! [C5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),A4)
           => ( B4 != aa_set_nat_set_nat(image_nat_nat(F),C5) ) ) ) ).

% subset_imageE
tff(fact_2617_subset__imageE,axiom,
    ! [B4: set_nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_int_set_nat(image_int_nat(F),A4))
     => ~ ! [C5: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(C5),A4)
           => ( B4 != aa_set_int_set_nat(image_int_nat(F),C5) ) ) ) ).

% subset_imageE
tff(fact_2618_subset__imageE,axiom,
    ! [B4: set_int,F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),aa_set_nat_set_int(image_nat_int(F),A4))
     => ~ ! [C5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),A4)
           => ( B4 != aa_set_nat_set_int(image_nat_int(F),C5) ) ) ) ).

% subset_imageE
tff(fact_2619_subset__imageE,axiom,
    ! [B4: set_int,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),aa_set_int_set_int(image_int_int(F),A4))
     => ~ ! [C5: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(C5),A4)
           => ( B4 != aa_set_int_set_int(image_int_int(F),C5) ) ) ) ).

% subset_imageE
tff(fact_2620_image__subset__iff,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)),B4)
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X5)),B4) ) ) ).

% image_subset_iff
tff(fact_2621_image__subset__iff,axiom,
    ! [F: fun_int_nat,A4: set_int,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_int_set_nat(image_int_nat(F),A4)),B4)
    <=> ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_set_nat_o(member_nat(aa_int_nat(F,X5)),B4) ) ) ).

% image_subset_iff
tff(fact_2622_image__subset__iff,axiom,
    ! [F: fun_nat_set_nat,A4: set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4)),B4)
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_set_set_nat_o(member_set_nat(aa_nat_set_nat(F,X5)),B4) ) ) ).

% image_subset_iff
tff(fact_2623_image__subset__iff,axiom,
    ! [F: fun_nat_int,A4: set_nat,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B4)
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_set_int_o(member_int(aa_nat_int(F,X5)),B4) ) ) ).

% image_subset_iff
tff(fact_2624_image__subset__iff,axiom,
    ! [F: fun_int_int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B4)
    <=> ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_set_int_o(member_int(aa_int_int(F,X5)),B4) ) ) ).

% image_subset_iff
tff(fact_2625_subset__image__iff,axiom,
    ! [B4: set_set_nat,F: fun_nat_set_nat,A4: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B4),aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,AA),A4)
          & ( B4 = aa_set7246812951530217472et_nat(image_nat_set_nat(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2626_subset__image__iff,axiom,
    ! [B4: set_nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(image_nat_nat(F),A4))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,AA),A4)
          & ( B4 = aa_set_nat_set_nat(image_nat_nat(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2627_subset__image__iff,axiom,
    ! [B4: set_nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_int_set_nat(image_int_nat(F),A4))
    <=> ? [AA: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(AA),A4)
          & ( B4 = aa_set_int_set_nat(image_int_nat(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2628_subset__image__iff,axiom,
    ! [B4: set_int,F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),aa_set_nat_set_int(image_nat_int(F),A4))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,AA),A4)
          & ( B4 = aa_set_nat_set_int(image_nat_int(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2629_subset__image__iff,axiom,
    ! [B4: set_int,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),aa_set_int_set_int(image_int_int(F),A4))
    <=> ? [AA: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(AA),A4)
          & ( B4 = aa_set_int_set_int(image_int_int(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_2630_all__subset__image,axiom,
    ! [F: fun_nat_set_nat,A4: set_nat,P2: fun_set_set_nat_o] :
      ( ! [B8: set_set_nat] :
          ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B8),aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4))
         => aa_set_set_nat_o(P2,B8) )
    <=> ! [B8: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),A4)
         => aa_set_set_nat_o(P2,aa_set7246812951530217472et_nat(image_nat_set_nat(F),B8)) ) ) ).

% all_subset_image
tff(fact_2631_all__subset__image,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( ! [B8: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),aa_set_nat_set_nat(image_nat_nat(F),A4))
         => aa_set_nat_o(P2,B8) )
    <=> ! [B8: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),A4)
         => aa_set_nat_o(P2,aa_set_nat_set_nat(image_nat_nat(F),B8)) ) ) ).

% all_subset_image
tff(fact_2632_all__subset__image,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_set_nat_o] :
      ( ! [B8: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),aa_set_int_set_nat(image_int_nat(F),A4))
         => aa_set_nat_o(P2,B8) )
    <=> ! [B8: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(B8),A4)
         => aa_set_nat_o(P2,aa_set_int_set_nat(image_int_nat(F),B8)) ) ) ).

% all_subset_image
tff(fact_2633_all__subset__image,axiom,
    ! [F: fun_nat_int,A4: set_nat,P2: fun_set_int_o] :
      ( ! [B8: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(B8),aa_set_nat_set_int(image_nat_int(F),A4))
         => aa_set_int_o(P2,B8) )
    <=> ! [B8: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),A4)
         => aa_set_int_o(P2,aa_set_nat_set_int(image_nat_int(F),B8)) ) ) ).

% all_subset_image
tff(fact_2634_all__subset__image,axiom,
    ! [F: fun_int_int,A4: set_int,P2: fun_set_int_o] :
      ( ! [B8: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(B8),aa_set_int_set_int(image_int_int(F),A4))
         => aa_set_int_o(P2,B8) )
    <=> ! [B8: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(B8),A4)
         => aa_set_int_o(P2,aa_set_int_set_int(image_int_int(F),B8)) ) ) ).

% all_subset_image
tff(fact_2635_verit__sum__simplify,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(plus_plus_uint32(A),zero_zero_uint32) = A ) ).

% verit_sum_simplify
tff(fact_2636_verit__sum__simplify,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ) ).

% verit_sum_simplify
tff(fact_2637_verit__sum__simplify,axiom,
    ! [A: rat] : ( aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ) ).

% verit_sum_simplify
tff(fact_2638_verit__sum__simplify,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ) ).

% verit_sum_simplify
tff(fact_2639_verit__sum__simplify,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ) ).

% verit_sum_simplify
tff(fact_2640_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(plus_plus_uint32(zero_zero_uint32),A) = A ) ).

% comm_monoid_add_class.add_0
tff(fact_2641_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ) ).

% comm_monoid_add_class.add_0
tff(fact_2642_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : ( aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ) ).

% comm_monoid_add_class.add_0
tff(fact_2643_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ) ).

% comm_monoid_add_class.add_0
tff(fact_2644_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ) ).

% comm_monoid_add_class.add_0
tff(fact_2645_add_Ocomm__neutral,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(plus_plus_uint32(A),zero_zero_uint32) = A ) ).

% add.comm_neutral
tff(fact_2646_add_Ocomm__neutral,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ) ).

% add.comm_neutral
tff(fact_2647_add_Ocomm__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ) ).

% add.comm_neutral
tff(fact_2648_add_Ocomm__neutral,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ) ).

% add.comm_neutral
tff(fact_2649_add_Ocomm__neutral,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ) ).

% add.comm_neutral
tff(fact_2650_add_Ogroup__left__neutral,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(plus_plus_uint32(zero_zero_uint32),A) = A ) ).

% add.group_left_neutral
tff(fact_2651_add_Ogroup__left__neutral,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ) ).

% add.group_left_neutral
tff(fact_2652_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ) ).

% add.group_left_neutral
tff(fact_2653_add_Ogroup__left__neutral,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ) ).

% add.group_left_neutral
tff(fact_2654_add__le__imp__le__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_2655_add__le__imp__le__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2656_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2657_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_2658_add__le__imp__le__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_2659_add__le__imp__le__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2660_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2661_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_2662_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C6: nat] : ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C6) ) ) ).

% le_iff_add
tff(fact_2663_add__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ) ).

% add_right_mono
tff(fact_2664_add__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)) ) ).

% add_right_mono
tff(fact_2665_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% add_right_mono
tff(fact_2666_add__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% add_right_mono
tff(fact_2667_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C: nat] : ( B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ) ) ).

% less_eqE
tff(fact_2668_add__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) ) ).

% add_left_mono
tff(fact_2669_add__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)) ) ).

% add_left_mono
tff(fact_2670_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) ) ).

% add_left_mono
tff(fact_2671_add__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% add_left_mono
tff(fact_2672_add__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_2673_add__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_mono
tff(fact_2674_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_2675_add__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_2676_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2677_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2678_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2679_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2680_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2681_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2682_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2683_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2684_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2685_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2686_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2687_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2688_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2689_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2690_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2691_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2692_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2693_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2694_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2695_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2696_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2697_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2698_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2699_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2700_add__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_2701_add__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_strict_mono
tff(fact_2702_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_2703_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_2704_add__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) ) ).

% add_strict_left_mono
tff(fact_2705_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B)) ) ).

% add_strict_left_mono
tff(fact_2706_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)) ) ).

% add_strict_left_mono
tff(fact_2707_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% add_strict_left_mono
tff(fact_2708_add__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ) ).

% add_strict_right_mono
tff(fact_2709_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2)) ) ).

% add_strict_right_mono
tff(fact_2710_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% add_strict_right_mono
tff(fact_2711_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% add_strict_right_mono
tff(fact_2712_add__less__imp__less__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_2713_add__less__imp__less__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A)),aa_rat_rat(plus_plus_rat(C2),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_2714_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_2715_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_2716_add__less__imp__less__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_2717_add__less__imp__less__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_2718_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_2719_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_2720_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : ( aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) ) ).

% diff_diff_eq
tff(fact_2721_diff__diff__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32] : ( minus_minus_uint32(minus_minus_uint32(A,B),C2) = minus_minus_uint32(A,aa_uint32_uint32(plus_plus_uint32(B),C2)) ) ).

% diff_diff_eq
tff(fact_2722_diff__diff__eq,axiom,
    ! [A: real,B: real,C2: real] : ( aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C2) = aa_real_real(minus_minus_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) ) ).

% diff_diff_eq
tff(fact_2723_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)) ) ).

% diff_diff_eq
tff(fact_2724_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) ) ).

% diff_diff_eq
tff(fact_2725_add__implies__diff,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C2),B) = A )
     => ( C2 = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_2726_add__implies__diff,axiom,
    ! [C2: uint32,B: uint32,A: uint32] :
      ( ( aa_uint32_uint32(plus_plus_uint32(C2),B) = A )
     => ( C2 = minus_minus_uint32(A,B) ) ) ).

% add_implies_diff
tff(fact_2727_add__implies__diff,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B) = A )
     => ( C2 = aa_real_real(minus_minus_real(A),B) ) ) ).

% add_implies_diff
tff(fact_2728_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B) = A )
     => ( C2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_2729_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) = A )
     => ( C2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_2730_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C2: rat] : ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C2)),B) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_2731_diff__add__eq__diff__diff__swap,axiom,
    ! [A: uint32,B: uint32,C2: uint32] : ( minus_minus_uint32(A,aa_uint32_uint32(plus_plus_uint32(B),C2)) = minus_minus_uint32(minus_minus_uint32(A,C2),B) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_2732_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C2: real] : ( aa_real_real(minus_minus_real(A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C2)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C2)),B) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_2733_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C2)),B) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_2734_diff__add__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),B) ) ).

% diff_add_eq
tff(fact_2735_diff__add__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32] : ( aa_uint32_uint32(plus_plus_uint32(minus_minus_uint32(A,B)),C2) = minus_minus_uint32(aa_uint32_uint32(plus_plus_uint32(A),C2),B) ) ).

% diff_add_eq
tff(fact_2736_diff__add__eq,axiom,
    ! [A: real,B: real,C2: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A),B)),C2) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ) ).

% diff_add_eq
tff(fact_2737_diff__add__eq,axiom,
    ! [A: int,B: int,C2: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ) ).

% diff_add_eq
tff(fact_2738_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C2: rat] : ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C2)),B) ) ).

% diff_diff_eq2
tff(fact_2739_diff__diff__eq2,axiom,
    ! [A: uint32,B: uint32,C2: uint32] : ( minus_minus_uint32(A,minus_minus_uint32(B,C2)) = minus_minus_uint32(aa_uint32_uint32(plus_plus_uint32(A),C2),B) ) ).

% diff_diff_eq2
tff(fact_2740_diff__diff__eq2,axiom,
    ! [A: real,B: real,C2: real] : ( aa_real_real(minus_minus_real(A),aa_real_real(minus_minus_real(B),C2)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ) ).

% diff_diff_eq2
tff(fact_2741_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ) ).

% diff_diff_eq2
tff(fact_2742_add__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] : ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C2) ) ).

% add_diff_eq
tff(fact_2743_add__diff__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32] : ( aa_uint32_uint32(plus_plus_uint32(A),minus_minus_uint32(B,C2)) = minus_minus_uint32(aa_uint32_uint32(plus_plus_uint32(A),B),C2) ) ).

% add_diff_eq
tff(fact_2744_add__diff__eq,axiom,
    ! [A: real,B: real,C2: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(minus_minus_real(B),C2)) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) ) ).

% add_diff_eq
tff(fact_2745_add__diff__eq,axiom,
    ! [A: int,B: int,C2: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ) ).

% add_diff_eq
tff(fact_2746_eq__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C2),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_2747_eq__diff__eq,axiom,
    ! [A: uint32,C2: uint32,B: uint32] :
      ( ( A = minus_minus_uint32(C2,B) )
    <=> ( aa_uint32_uint32(plus_plus_uint32(A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_2748_eq__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( A = aa_real_real(minus_minus_real(C2),B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_2749_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C2 ) ) ).

% eq_diff_eq
tff(fact_2750_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C2 )
    <=> ( A = aa_rat_rat(plus_plus_rat(C2),B) ) ) ).

% diff_eq_eq
tff(fact_2751_diff__eq__eq,axiom,
    ! [A: uint32,B: uint32,C2: uint32] :
      ( ( minus_minus_uint32(A,B) = C2 )
    <=> ( A = aa_uint32_uint32(plus_plus_uint32(C2),B) ) ) ).

% diff_eq_eq
tff(fact_2752_diff__eq__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = C2 )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B) ) ) ).

% diff_eq_eq
tff(fact_2753_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C2 )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B) ) ) ).

% diff_eq_eq
tff(fact_2754_group__cancel_Osub1,axiom,
    ! [A4: rat,K: rat,A: rat,B: rat] :
      ( ( A4 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(minus_minus_rat(A4),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_2755_group__cancel_Osub1,axiom,
    ! [A4: uint32,K: uint32,A: uint32,B: uint32] :
      ( ( A4 = aa_uint32_uint32(plus_plus_uint32(K),A) )
     => ( minus_minus_uint32(A4,B) = aa_uint32_uint32(plus_plus_uint32(K),minus_minus_uint32(A,B)) ) ) ).

% group_cancel.sub1
tff(fact_2756_group__cancel_Osub1,axiom,
    ! [A4: real,K: real,A: real,B: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(minus_minus_real(A4),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_2757_group__cancel_Osub1,axiom,
    ! [A4: int,K: int,A: int,B: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_2758_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A: nat] :
      ( ( A4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_2759_add__Suc,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ) ).

% add_Suc
tff(fact_2760_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) ) ).

% add_Suc_shift
tff(fact_2761_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Na) = Na ) ).

% plus_nat.add_0
tff(fact_2762_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = M )
     => ( Na = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_2763_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_2764_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_2765_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_2766_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_2767_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_2768_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_less_add1
tff(fact_2769_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_less_add2
tff(fact_2770_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% less_add_eq_less
tff(fact_2771_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ? [K3: nat] : ( Na = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ) ).

% nat_le_iff_add
tff(fact_2772_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_le_add2
tff(fact_2773_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_le_add1
tff(fact_2774_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_2775_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_2776_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N: nat] : ( L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ) ).

% le_Suc_ex
tff(fact_2777_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% add_leD2
tff(fact_2778_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% add_leD1
tff(fact_2779_le__add2,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% le_add2
tff(fact_2780_le__add1,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) ).

% le_add1
tff(fact_2781_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% add_leE
tff(fact_2782_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),Na) = M ) ).

% diff_add_inverse2
tff(fact_2783_diff__add__inverse,axiom,
    ! [Na: 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,Na),M)),Na) = M ) ).

% diff_add_inverse
tff(fact_2784_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: 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,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ).

% diff_cancel2
tff(fact_2785_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ).

% Nat.diff_cancel
tff(fact_2786_enumerate__in__set,axiom,
    ! [S2: set_nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => aa_set_nat_o(member_nat(aa_nat_nat(infini8530281810654367211te_nat(S2),Na)),S2) ) ).

% enumerate_in_set
tff(fact_2787_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( aa_set_real_o(member_real(A),ring_1_Ints_real)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2788_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
     => ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2789_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( aa_set_int_o(member_int(A),ring_1_Ints_int)
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2790_enumerate__Ex,axiom,
    ! [S2: set_nat,Sa: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(member_nat(Sa),S2)
       => ? [N: nat] : ( aa_nat_nat(infini8530281810654367211te_nat(S2),N) = Sa ) ) ) ).

% enumerate_Ex
tff(fact_2791_all__finite__subset__image,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( ! [B8: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B8)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),aa_set_nat_set_nat(image_nat_nat(F),A4)) )
         => aa_set_nat_o(P2,B8) )
    <=> ! [B8: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B8)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),A4) )
         => aa_set_nat_o(P2,aa_set_nat_set_nat(image_nat_nat(F),B8)) ) ) ).

% all_finite_subset_image
tff(fact_2792_all__finite__subset__image,axiom,
    ! [F: fun_complex_nat,A4: set_complex,P2: fun_set_nat_o] :
      ( ! [B8: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B8)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),image_complex_nat(F,A4)) )
         => aa_set_nat_o(P2,B8) )
    <=> ! [B8: set_complex] :
          ( ( aa_set_complex_o(finite3207457112153483333omplex,B8)
            & aa_set_complex_o(ord_le211207098394363844omplex(B8),A4) )
         => aa_set_nat_o(P2,image_complex_nat(F,B8)) ) ) ).

% all_finite_subset_image
tff(fact_2793_all__finite__subset__image,axiom,
    ! [F: fun_Code_integer_nat,A4: set_Code_integer,P2: fun_set_nat_o] :
      ( ! [B8: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B8)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),image_951025933927791156er_nat(F,A4)) )
         => aa_set_nat_o(P2,B8) )
    <=> ! [B8: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),A4) )
         => aa_set_nat_o(P2,image_951025933927791156er_nat(F,B8)) ) ) ).

% all_finite_subset_image
tff(fact_2794_all__finite__subset__image,axiom,
    ! [F: fun_nat_complex,A4: set_nat,P2: fun_set_complex_o] :
      ( ! [B8: set_complex] :
          ( ( aa_set_complex_o(finite3207457112153483333omplex,B8)
            & aa_set_complex_o(ord_le211207098394363844omplex(B8),image_nat_complex(F,A4)) )
         => aa_set_complex_o(P2,B8) )
    <=> ! [B8: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B8)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),A4) )
         => aa_set_complex_o(P2,image_nat_complex(F,B8)) ) ) ).

% all_finite_subset_image
tff(fact_2795_all__finite__subset__image,axiom,
    ! [F: fun_complex_complex,A4: set_complex,P2: fun_set_complex_o] :
      ( ! [B8: set_complex] :
          ( ( aa_set_complex_o(finite3207457112153483333omplex,B8)
            & aa_set_complex_o(ord_le211207098394363844omplex(B8),image_1468599708987790691omplex(F,A4)) )
         => aa_set_complex_o(P2,B8) )
    <=> ! [B8: set_complex] :
          ( ( aa_set_complex_o(finite3207457112153483333omplex,B8)
            & aa_set_complex_o(ord_le211207098394363844omplex(B8),A4) )
         => aa_set_complex_o(P2,image_1468599708987790691omplex(F,B8)) ) ) ).

% all_finite_subset_image
tff(fact_2796_all__finite__subset__image,axiom,
    ! [F: fun_Co1900716531650368264omplex,A4: set_Code_integer,P2: fun_set_complex_o] :
      ( ! [B8: set_complex] :
          ( ( aa_set_complex_o(finite3207457112153483333omplex,B8)
            & aa_set_complex_o(ord_le211207098394363844omplex(B8),image_3397630267976458002omplex(F,A4)) )
         => aa_set_complex_o(P2,B8) )
    <=> ! [B8: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),A4) )
         => aa_set_complex_o(P2,image_3397630267976458002omplex(F,B8)) ) ) ).

% all_finite_subset_image
tff(fact_2797_all__finite__subset__image,axiom,
    ! [F: fun_nat_Code_integer,A4: set_nat,P2: fun_se7289222521248229150eger_o] :
      ( ! [B8: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),image_1215581382706833972nteger(F,A4)) )
         => aa_set5643387332196575111eger_o(P2,B8) )
    <=> ! [B8: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B8)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),A4) )
         => aa_set5643387332196575111eger_o(P2,image_1215581382706833972nteger(F,B8)) ) ) ).

% all_finite_subset_image
tff(fact_2798_all__finite__subset__image,axiom,
    ! [F: fun_co8153928105989009288nteger,A4: set_complex,P2: fun_se7289222521248229150eger_o] :
      ( ! [B8: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),image_1994230757181692690nteger(F,A4)) )
         => aa_set5643387332196575111eger_o(P2,B8) )
    <=> ! [B8: set_complex] :
          ( ( aa_set_complex_o(finite3207457112153483333omplex,B8)
            & aa_set_complex_o(ord_le211207098394363844omplex(B8),A4) )
         => aa_set5643387332196575111eger_o(P2,image_1994230757181692690nteger(F,B8)) ) ) ).

% all_finite_subset_image
tff(fact_2799_all__finite__subset__image,axiom,
    ! [F: fun_Co8499238673658658103nteger,A4: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( ! [B8: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),image_4470545334726330049nteger(F,A4)) )
         => aa_set5643387332196575111eger_o(P2,B8) )
    <=> ! [B8: set_Code_integer] :
          ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
            & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),A4) )
         => aa_set5643387332196575111eger_o(P2,image_4470545334726330049nteger(F,B8)) ) ) ).

% all_finite_subset_image
tff(fact_2800_all__finite__subset__image,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_set_nat_o] :
      ( ! [B8: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B8)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),aa_set_int_set_nat(image_int_nat(F),A4)) )
         => aa_set_nat_o(P2,B8) )
    <=> ! [B8: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B8)
            & aa_set_int_o(ord_less_eq_set_int(B8),A4) )
         => aa_set_nat_o(P2,aa_set_int_set_nat(image_int_nat(F),B8)) ) ) ).

% all_finite_subset_image
tff(fact_2801_ex__finite__subset__image,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( ? [B8: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B8)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),aa_set_nat_set_nat(image_nat_nat(F),A4))
          & aa_set_nat_o(P2,B8) )
    <=> ? [B8: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B8)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),A4)
          & aa_set_nat_o(P2,aa_set_nat_set_nat(image_nat_nat(F),B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2802_ex__finite__subset__image,axiom,
    ! [F: fun_complex_nat,A4: set_complex,P2: fun_set_nat_o] :
      ( ? [B8: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B8)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),image_complex_nat(F,A4))
          & aa_set_nat_o(P2,B8) )
    <=> ? [B8: set_complex] :
          ( aa_set_complex_o(finite3207457112153483333omplex,B8)
          & aa_set_complex_o(ord_le211207098394363844omplex(B8),A4)
          & aa_set_nat_o(P2,image_complex_nat(F,B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2803_ex__finite__subset__image,axiom,
    ! [F: fun_Code_integer_nat,A4: set_Code_integer,P2: fun_set_nat_o] :
      ( ? [B8: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B8)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),image_951025933927791156er_nat(F,A4))
          & aa_set_nat_o(P2,B8) )
    <=> ? [B8: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),A4)
          & aa_set_nat_o(P2,image_951025933927791156er_nat(F,B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2804_ex__finite__subset__image,axiom,
    ! [F: fun_nat_complex,A4: set_nat,P2: fun_set_complex_o] :
      ( ? [B8: set_complex] :
          ( aa_set_complex_o(finite3207457112153483333omplex,B8)
          & aa_set_complex_o(ord_le211207098394363844omplex(B8),image_nat_complex(F,A4))
          & aa_set_complex_o(P2,B8) )
    <=> ? [B8: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B8)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),A4)
          & aa_set_complex_o(P2,image_nat_complex(F,B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2805_ex__finite__subset__image,axiom,
    ! [F: fun_complex_complex,A4: set_complex,P2: fun_set_complex_o] :
      ( ? [B8: set_complex] :
          ( aa_set_complex_o(finite3207457112153483333omplex,B8)
          & aa_set_complex_o(ord_le211207098394363844omplex(B8),image_1468599708987790691omplex(F,A4))
          & aa_set_complex_o(P2,B8) )
    <=> ? [B8: set_complex] :
          ( aa_set_complex_o(finite3207457112153483333omplex,B8)
          & aa_set_complex_o(ord_le211207098394363844omplex(B8),A4)
          & aa_set_complex_o(P2,image_1468599708987790691omplex(F,B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2806_ex__finite__subset__image,axiom,
    ! [F: fun_Co1900716531650368264omplex,A4: set_Code_integer,P2: fun_set_complex_o] :
      ( ? [B8: set_complex] :
          ( aa_set_complex_o(finite3207457112153483333omplex,B8)
          & aa_set_complex_o(ord_le211207098394363844omplex(B8),image_3397630267976458002omplex(F,A4))
          & aa_set_complex_o(P2,B8) )
    <=> ? [B8: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),A4)
          & aa_set_complex_o(P2,image_3397630267976458002omplex(F,B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2807_ex__finite__subset__image,axiom,
    ! [F: fun_nat_Code_integer,A4: set_nat,P2: fun_se7289222521248229150eger_o] :
      ( ? [B8: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),image_1215581382706833972nteger(F,A4))
          & aa_set5643387332196575111eger_o(P2,B8) )
    <=> ? [B8: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B8)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),A4)
          & aa_set5643387332196575111eger_o(P2,image_1215581382706833972nteger(F,B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2808_ex__finite__subset__image,axiom,
    ! [F: fun_co8153928105989009288nteger,A4: set_complex,P2: fun_se7289222521248229150eger_o] :
      ( ? [B8: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),image_1994230757181692690nteger(F,A4))
          & aa_set5643387332196575111eger_o(P2,B8) )
    <=> ? [B8: set_complex] :
          ( aa_set_complex_o(finite3207457112153483333omplex,B8)
          & aa_set_complex_o(ord_le211207098394363844omplex(B8),A4)
          & aa_set5643387332196575111eger_o(P2,image_1994230757181692690nteger(F,B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2809_ex__finite__subset__image,axiom,
    ! [F: fun_Co8499238673658658103nteger,A4: set_Code_integer,P2: fun_se7289222521248229150eger_o] :
      ( ? [B8: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),image_4470545334726330049nteger(F,A4))
          & aa_set5643387332196575111eger_o(P2,B8) )
    <=> ? [B8: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8)
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B8),A4)
          & aa_set5643387332196575111eger_o(P2,image_4470545334726330049nteger(F,B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2810_ex__finite__subset__image,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_set_nat_o] :
      ( ? [B8: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B8)
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B8),aa_set_int_set_nat(image_int_nat(F),A4))
          & aa_set_nat_o(P2,B8) )
    <=> ? [B8: set_int] :
          ( aa_set_int_o(finite_finite_int,B8)
          & aa_set_int_o(ord_less_eq_set_int(B8),A4)
          & aa_set_nat_o(P2,aa_set_int_set_nat(image_int_nat(F),B8)) ) ) ).

% ex_finite_subset_image
tff(fact_2811_finite__subset__image,axiom,
    ! [B4: set_nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ? [C5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),A4)
            & aa_set_nat_o(finite_finite_nat,C5)
            & ( B4 = aa_set_nat_set_nat(image_nat_nat(F),C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2812_finite__subset__image,axiom,
    ! [B4: set_nat,F: fun_complex_nat,A4: set_complex] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),image_complex_nat(F,A4))
       => ? [C5: set_complex] :
            ( aa_set_complex_o(ord_le211207098394363844omplex(C5),A4)
            & aa_set_complex_o(finite3207457112153483333omplex,C5)
            & ( B4 = image_complex_nat(F,C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2813_finite__subset__image,axiom,
    ! [B4: set_nat,F: fun_Code_integer_nat,A4: set_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),image_951025933927791156er_nat(F,A4))
       => ? [C5: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(C5),A4)
            & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
            & ( B4 = image_951025933927791156er_nat(F,C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2814_finite__subset__image,axiom,
    ! [B4: set_complex,F: fun_nat_complex,A4: set_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_nat_complex(F,A4))
       => ? [C5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),A4)
            & aa_set_nat_o(finite_finite_nat,C5)
            & ( B4 = image_nat_complex(F,C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2815_finite__subset__image,axiom,
    ! [B4: set_complex,F: fun_complex_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_1468599708987790691omplex(F,A4))
       => ? [C5: set_complex] :
            ( aa_set_complex_o(ord_le211207098394363844omplex(C5),A4)
            & aa_set_complex_o(finite3207457112153483333omplex,C5)
            & ( B4 = image_1468599708987790691omplex(F,C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2816_finite__subset__image,axiom,
    ! [B4: set_complex,F: fun_Co1900716531650368264omplex,A4: set_Code_integer] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_3397630267976458002omplex(F,A4))
       => ? [C5: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(C5),A4)
            & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
            & ( B4 = image_3397630267976458002omplex(F,C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2817_finite__subset__image,axiom,
    ! [B4: set_Code_integer,F: fun_nat_Code_integer,A4: set_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),image_1215581382706833972nteger(F,A4))
       => ? [C5: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C5),A4)
            & aa_set_nat_o(finite_finite_nat,C5)
            & ( B4 = image_1215581382706833972nteger(F,C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2818_finite__subset__image,axiom,
    ! [B4: set_Code_integer,F: fun_co8153928105989009288nteger,A4: set_complex] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),image_1994230757181692690nteger(F,A4))
       => ? [C5: set_complex] :
            ( aa_set_complex_o(ord_le211207098394363844omplex(C5),A4)
            & aa_set_complex_o(finite3207457112153483333omplex,C5)
            & ( B4 = image_1994230757181692690nteger(F,C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2819_finite__subset__image,axiom,
    ! [B4: set_Code_integer,F: fun_Co8499238673658658103nteger,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),image_4470545334726330049nteger(F,A4))
       => ? [C5: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(C5),A4)
            & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C5)
            & ( B4 = image_4470545334726330049nteger(F,C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2820_finite__subset__image,axiom,
    ! [B4: set_nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_int_set_nat(image_int_nat(F),A4))
       => ? [C5: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(C5),A4)
            & aa_set_int_o(finite_finite_int,C5)
            & ( B4 = aa_set_int_set_nat(image_int_nat(F),C5) ) ) ) ) ).

% finite_subset_image
tff(fact_2821_finite__surj,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => aa_set_nat_o(finite_finite_nat,B4) ) ) ).

% finite_surj
tff(fact_2822_finite__surj,axiom,
    ! [A4: set_nat,B4: set_complex,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_nat_complex(F,A4))
       => aa_set_complex_o(finite3207457112153483333omplex,B4) ) ) ).

% finite_surj
tff(fact_2823_finite__surj,axiom,
    ! [A4: set_nat,B4: set_Code_integer,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),image_1215581382706833972nteger(F,A4))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4) ) ) ).

% finite_surj
tff(fact_2824_finite__surj,axiom,
    ! [A4: set_int,B4: set_nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_int_set_nat(image_int_nat(F),A4))
       => aa_set_nat_o(finite_finite_nat,B4) ) ) ).

% finite_surj
tff(fact_2825_finite__surj,axiom,
    ! [A4: set_int,B4: set_complex,F: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_int_complex(F,A4))
       => aa_set_complex_o(finite3207457112153483333omplex,B4) ) ) ).

% finite_surj
tff(fact_2826_finite__surj,axiom,
    ! [A4: set_int,B4: set_Code_integer,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),image_1587234942943678608nteger(F,A4))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4) ) ) ).

% finite_surj
tff(fact_2827_finite__surj,axiom,
    ! [A4: set_complex,B4: set_nat,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),image_complex_nat(F,A4))
       => aa_set_nat_o(finite_finite_nat,B4) ) ) ).

% finite_surj
tff(fact_2828_finite__surj,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_1468599708987790691omplex(F,A4))
       => aa_set_complex_o(finite3207457112153483333omplex,B4) ) ) ).

% finite_surj
tff(fact_2829_finite__surj,axiom,
    ! [A4: set_complex,B4: set_Code_integer,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),image_1994230757181692690nteger(F,A4))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4) ) ) ).

% finite_surj
tff(fact_2830_finite__surj,axiom,
    ! [A4: set_Code_integer,B4: set_nat,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),image_951025933927791156er_nat(F,A4))
       => aa_set_nat_o(finite_finite_nat,B4) ) ) ).

% finite_surj
tff(fact_2831_image__diff__subset,axiom,
    ! [F: fun_nat_set_nat,A4: set_nat,B4: set_nat] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4),aa_set7246812951530217472et_nat(image_nat_set_nat(F),B4))),aa_set7246812951530217472et_nat(image_nat_set_nat(F),minus_minus_set_nat(A4,B4))) ).

% image_diff_subset
tff(fact_2832_image__diff__subset,axiom,
    ! [F: fun_int_nat,A4: set_int,B4: set_int] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(aa_set_int_set_nat(image_int_nat(F),A4),aa_set_int_set_nat(image_int_nat(F),B4))),aa_set_int_set_nat(image_int_nat(F),minus_minus_set_int(A4,B4))) ).

% image_diff_subset
tff(fact_2833_image__diff__subset,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4),aa_set_nat_set_nat(image_nat_nat(F),B4))),aa_set_nat_set_nat(image_nat_nat(F),minus_minus_set_nat(A4,B4))) ).

% image_diff_subset
tff(fact_2834_image__diff__subset,axiom,
    ! [F: fun_int_int,A4: set_int,B4: set_int] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(aa_set_int_set_int(image_int_int(F),A4),aa_set_int_set_int(image_int_int(F),B4))),aa_set_int_set_int(image_int_int(F),minus_minus_set_int(A4,B4))) ).

% image_diff_subset
tff(fact_2835_image__diff__subset,axiom,
    ! [F: fun_nat_int,A4: set_nat,B4: set_nat] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set_nat_set_int(image_nat_int(F),B4))),aa_set_nat_set_int(image_nat_int(F),minus_minus_set_nat(A4,B4))) ).

% image_diff_subset
tff(fact_2836_add__nonpos__eq__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y) = zero_zero_real )
        <=> ( ( Xa = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2837_add__nonpos__eq__0__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(Xa),Y) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2838_add__nonpos__eq__0__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2839_add__nonpos__eq__0__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2840_add__nonneg__eq__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y) = zero_zero_real )
        <=> ( ( Xa = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2841_add__nonneg__eq__0__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(plus_plus_rat(Xa),Y) = zero_zero_rat )
        <=> ( ( Xa = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2842_add__nonneg__eq__0__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = zero_zero_nat )
        <=> ( ( Xa = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2843_add__nonneg__eq__0__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y) = zero_zero_int )
        <=> ( ( Xa = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2844_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_2845_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_2846_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_2847_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_2848_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2849_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2850_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2851_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2852_add__increasing2,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_increasing2
tff(fact_2853_add__increasing2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% add_increasing2
tff(fact_2854_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_increasing2
tff(fact_2855_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_increasing2
tff(fact_2856_add__decreasing2,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_2857_add__decreasing2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_2858_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_2859_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ) ) ).

% add_decreasing2
tff(fact_2860_add__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_increasing
tff(fact_2861_add__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% add_increasing
tff(fact_2862_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_increasing
tff(fact_2863_add__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_increasing
tff(fact_2864_add__decreasing,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),B) ) ) ).

% add_decreasing
tff(fact_2865_add__decreasing,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C2)),B) ) ) ).

% add_decreasing
tff(fact_2866_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),B) ) ) ).

% add_decreasing
tff(fact_2867_add__decreasing,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),B) ) ) ).

% add_decreasing
tff(fact_2868_pos__add__strict,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% pos_add_strict
tff(fact_2869_pos__add__strict,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% pos_add_strict
tff(fact_2870_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% pos_add_strict
tff(fact_2871_pos__add__strict,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% pos_add_strict
tff(fact_2872_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_2873_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_2874_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_pos
tff(fact_2875_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_2876_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_2877_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_2878_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_2879_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_2880_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_2881_add__less__zeroD,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_2882_add__less__zeroD,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xa),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_2883_add__less__zeroD,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_2884_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2885_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2886_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2887_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2888_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2889_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2890_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2891_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2892_add__le__less__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2893_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_le_less_mono
tff(fact_2894_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2895_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2896_add__less__le__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2897_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C2)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_less_le_mono
tff(fact_2898_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2899_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2900_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real)) ) ).

% add_mono1
tff(fact_2901_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).

% add_mono1
tff(fact_2902_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_2903_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_2904_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_2905_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat)) ).

% less_add_one
tff(fact_2906_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_2907_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_2908_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C2 )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_2909_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_2910_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_2911_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_2912_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_2913_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_2914_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_2915_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_2916_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C2)),A)) ) ).

% le_add_diff
tff(fact_2917_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_2918_le__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(minus_minus_real(C2),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) ) ).

% le_diff_eq
tff(fact_2919_le__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C2) ) ).

% le_diff_eq
tff(fact_2920_le__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ) ).

% le_diff_eq
tff(fact_2921_diff__le__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),C2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) ) ).

% diff_le_eq
tff(fact_2922_diff__le__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C2),B)) ) ).

% diff_le_eq
tff(fact_2923_diff__le__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% diff_le_eq
tff(fact_2924_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,Na: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2925_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,Na: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(plus_plus_rat(J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),aa_rat_rat(plus_plus_rat(J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2926_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,Na: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2927_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,Na: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2928_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(minus_minus_real(Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_2929_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_2930_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_2931_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_2932_diff__less__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),C2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B)) ) ).

% diff_less_eq
tff(fact_2933_diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C2),B)) ) ).

% diff_less_eq
tff(fact_2934_diff__less__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C2),B)) ) ).

% diff_less_eq
tff(fact_2935_less__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(minus_minus_real(C2),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C2) ) ).

% less_diff_eq
tff(fact_2936_less__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C2),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C2) ) ).

% less_diff_eq
tff(fact_2937_less__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C2),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C2) ) ).

% less_diff_eq
tff(fact_2938_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2939_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2940_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2941_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2942_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( aa_set_real_o(member_real(A),ring_1_Ints_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A) != zero_zero_real ) ) ).

% Ints_odd_nonzero
tff(fact_2943_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(one_one_rat),A)),A) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_2944_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( aa_set_int_o(member_int(A),ring_1_Ints_int)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_2945_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_2946_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_2947_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ~ ! [Q5: nat] : ( Na != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q5)) ) ) ).

% less_natE
tff(fact_2948_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M))) ).

% less_add_Suc1
tff(fact_2949_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I))) ).

% less_add_Suc2
tff(fact_2950_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ? [K3: nat] : ( Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ) ).

% less_iff_Suc_add
tff(fact_2951_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ? [K2: nat] : ( Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ) ).

% less_imp_Suc_add
tff(fact_2952_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_2953_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : ( aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Na) ) ).

% Suc_eq_plus1_left
tff(fact_2954_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_2955_Suc__eq__plus1,axiom,
    ! [Na: nat] : ( aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat) ) ).

% Suc_eq_plus1
tff(fact_2956_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M3: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M3)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))) ) ).

% mono_nat_linear_lb
tff(fact_2957_diff__add__0,axiom,
    ! [Na: nat,M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = zero_zero_nat ) ).

% diff_add_0
tff(fact_2958_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_2959_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_2960_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_2961_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_2962_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_2963_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_2964_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_2965_le__enumerate,axiom,
    ! [S2: set_nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(infini8530281810654367211te_nat(S2),Na)) ) ).

% le_enumerate
tff(fact_2966_the__elem__image__unique,axiom,
    ! [A4: set_nat,F: fun_nat_set_nat,Xa: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),A4)
           => ( aa_nat_set_nat(F,Y3) = aa_nat_set_nat(F,Xa) ) )
       => ( the_elem_set_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4)) = aa_nat_set_nat(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_2967_the__elem__image__unique,axiom,
    ! [A4: set_nat,F: fun_nat_nat,Xa: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),A4)
           => ( aa_nat_nat(F,Y3) = aa_nat_nat(F,Xa) ) )
       => ( aa_set_nat_nat(the_elem_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) = aa_nat_nat(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_2968_the__elem__image__unique,axiom,
    ! [A4: set_nat,F: fun_nat_int,Xa: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [Y3: nat] :
            ( aa_set_nat_o(member_nat(Y3),A4)
           => ( aa_nat_int(F,Y3) = aa_nat_int(F,Xa) ) )
       => ( the_elem_int(aa_set_nat_set_int(image_nat_int(F),A4)) = aa_nat_int(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_2969_the__elem__image__unique,axiom,
    ! [A4: set_int,F: fun_int_nat,Xa: int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [Y3: int] :
            ( aa_set_int_o(member_int(Y3),A4)
           => ( aa_int_nat(F,Y3) = aa_int_nat(F,Xa) ) )
       => ( aa_set_nat_nat(the_elem_nat,aa_set_int_set_nat(image_int_nat(F),A4)) = aa_int_nat(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_2970_the__elem__image__unique,axiom,
    ! [A4: set_int,F: fun_int_int,Xa: int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [Y3: int] :
            ( aa_set_int_o(member_int(Y3),A4)
           => ( aa_int_int(F,Y3) = aa_int_int(F,Xa) ) )
       => ( the_elem_int(aa_set_int_set_int(image_int_int(F),A4)) = aa_int_int(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_2971_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( aa_set_real_o(member_real(A),ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% Ints_odd_less_0
tff(fact_2972_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( aa_set_rat_o(member_rat(A),ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(one_one_rat),A)),A)),zero_zero_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% Ints_odd_less_0
tff(fact_2973_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( aa_set_int_o(member_int(A),ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Ints_odd_less_0
tff(fact_2974_add__strict__increasing2,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2975_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2976_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2977_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2978_add__strict__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_2979_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C2)) ) ) ).

% add_strict_increasing
tff(fact_2980_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_2981_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C2)) ) ) ).

% add_strict_increasing
tff(fact_2982_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2983_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2984_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2985_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2986_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_2987_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_2988_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_2989_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_2990_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2991_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2992_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2993_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2994_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_2995_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_2996_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_2997_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_2998_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_2999_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_3000_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_3001_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_3002_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_3003_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_3004_nat__diff__split__asm,axiom,
    ! [P2: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P2,zero_zero_nat) )
          | ? [D5: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
              & ~ aa_nat_o(P2,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_3005_nat__diff__split,axiom,
    ! [P2: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P2,zero_zero_nat) )
        & ! [D5: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D5) )
           => aa_nat_o(P2,D5) ) ) ) ).

% nat_diff_split
tff(fact_3006_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_3007_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Summary),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))))) ) ).

% VEBT_internal.height.simps(2)
tff(fact_3008_enumerate__step,axiom,
    ! [S2: set_nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S2),Na)),aa_nat_nat(infini8530281810654367211te_nat(S2),aa_nat_nat(suc,Na))) ) ).

% enumerate_step
tff(fact_3009_enumerate__mono,axiom,
    ! [M: nat,Na: nat,S2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S2),M)),aa_nat_nat(infini8530281810654367211te_nat(S2),Na)) ) ) ).

% enumerate_mono
tff(fact_3010_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = $ite(M = zero_zero_nat,Na,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)),Na))) ) ).

% add_eq_if
tff(fact_3011_frac__add,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xa)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xa)),aa_real_real(archim2898591450579166408c_real,Y)),aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xa)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real)) ) ).

% frac_add
tff(fact_3012_frac__add,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(Xa),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,Xa)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,Xa)),aa_rat_rat(archimedean_frac_rat,Y)),aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,Xa)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat)) ) ).

% frac_add
tff(fact_3013_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Summary2),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_3014_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va2: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_3015_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,vEBT_Leaf((Uu2),(B))),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_3016_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,vEBT_Leaf((A),(Uw2))),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_3017_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% pred_bound_height'
tff(fact_3018_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% succ'_bound_height
tff(fact_3019_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% insert'_bound_height
tff(fact_3020_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))) ) ).

% member_bound_height'
tff(fact_3021_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = zero_zero_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf((A3),(B3))) ) )
         => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Summary2),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))))) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_3022_max__ins__scaled,axiom,
    ! [Na: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14)),image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,X13))))))) ).

% max_ins_scaled
tff(fact_3023_field__le__epsilon,axiom,
    ! [Xa: real,Y: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% field_le_epsilon
tff(fact_3024_field__le__epsilon,axiom,
    ! [Xa: rat,Y: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_rat_rat(plus_plus_rat(Y),E)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% field_le_epsilon
tff(fact_3025_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xa) = Y )
     => ( ! [A3: $o] :
            ( ? [B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((A3),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_3026_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((B3),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_3027_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((A3),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Leaf((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_3028_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((B3),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Leaf((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy),Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_3029_Euclid__induct,axiom,
    ! [P2: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3)
        <=> aa_nat_o(aa_nat_fun_nat_o(P2,B3),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,A3),zero_zero_nat)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3)
             => aa_nat_o(aa_nat_fun_nat_o(P2,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,A),B) ) ) ) ).

% Euclid_induct
tff(fact_3030_triangle__Suc,axiom,
    ! [Na: nat] : ( aa_nat_nat(nat_triangle,aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Na)),aa_nat_nat(suc,Na)) ) ).

% triangle_Suc
tff(fact_3031_mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_3032_mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_3033_mult__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_3034_mult__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_3035_mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_3036_mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_3037_mult__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_3038_mult__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
    <=> ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_3039_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_3040_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_3041_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_3042_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_3043_mult__zero__right,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ) ).

% mult_zero_right
tff(fact_3044_mult__zero__right,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),zero_zero_uint32) = zero_zero_uint32 ) ).

% mult_zero_right
tff(fact_3045_mult__zero__right,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ) ).

% mult_zero_right
tff(fact_3046_mult__zero__right,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ) ).

% mult_zero_right
tff(fact_3047_mult__zero__right,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ) ).

% mult_zero_right
tff(fact_3048_mult__zero__left,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ) ).

% mult_zero_left
tff(fact_3049_mult__zero__left,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,zero_zero_uint32),A) = zero_zero_uint32 ) ).

% mult_zero_left
tff(fact_3050_mult__zero__left,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ) ).

% mult_zero_left
tff(fact_3051_mult__zero__left,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ) ).

% mult_zero_left
tff(fact_3052_mult__zero__left,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ) ).

% mult_zero_left
tff(fact_3053_mult_Oright__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ) ).

% mult.right_neutral
tff(fact_3054_mult_Oright__neutral,axiom,
    ! [A: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ) ).

% mult.right_neutral
tff(fact_3055_mult_Oright__neutral,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),one_one_uint32) = A ) ).

% mult.right_neutral
tff(fact_3056_mult_Oright__neutral,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ) ).

% mult.right_neutral
tff(fact_3057_mult_Oright__neutral,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ) ).

% mult.right_neutral
tff(fact_3058_mult_Oright__neutral,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ) ).

% mult.right_neutral
tff(fact_3059_mult__1,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ) ).

% mult_1
tff(fact_3060_mult__1,axiom,
    ! [A: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ) ).

% mult_1
tff(fact_3061_mult__1,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,one_one_uint32),A) = A ) ).

% mult_1
tff(fact_3062_mult__1,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ) ).

% mult_1
tff(fact_3063_mult__1,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ) ).

% mult_1
tff(fact_3064_mult__1,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ) ).

% mult_1
tff(fact_3065_mult__cancel2,axiom,
    ! [M: nat,K: nat,Na: 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,Na),K) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_3066_mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: 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),Na) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_3067_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_3068_mult__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( Na = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_3069_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_3070_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_3071_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_3072_mult__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_3073_mult__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_3074_mult__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_3075_mult__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_3076_mult__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_3077_mult__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_3078_mult__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_3079_mult__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_3080_mult__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_3081_mult__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_3082_mult__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
    <=> ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_3083_mult__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( C2 = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
    <=> ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_3084_mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_3085_one__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_3086_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% mult_less_cancel2
tff(fact_3087_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_3088_mult__Suc__right,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ).

% mult_Suc_right
tff(fact_3089_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Na) ) ) ).

% one_le_mult_iff
tff(fact_3090_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% mult_le_cancel2
tff(fact_3091_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C2: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2)) ) ).

% mult.assoc
tff(fact_3092_mult_Oassoc,axiom,
    ! [A: uint32,B: uint32,C2: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),B)),C2) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,B),C2)) ) ).

% mult.assoc
tff(fact_3093_mult_Oassoc,axiom,
    ! [A: real,B: real,C2: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ).

% mult.assoc
tff(fact_3094_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ).

% mult.assoc
tff(fact_3095_mult_Oassoc,axiom,
    ! [A: int,B: int,C2: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ).

% mult.assoc
tff(fact_3096_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: assn,B: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3097_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: uint32,B: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),B) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3098_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: real,B: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3099_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3100_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_3101_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C2: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3102_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: uint32,A: uint32,C2: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,B),aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),C2)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,B),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3103_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3104_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3105_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_3106_mult__right__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3107_mult__right__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3108_mult__right__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3109_mult__right__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C2) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_3110_mult__left__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3111_mult__left__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3112_mult__left__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3113_mult__left__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C2),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_3114_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_3115_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_3116_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_3117_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_3118_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_3119_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_3120_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_3121_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_3122_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_3123_mult__not__zero,axiom,
    ! [A: uint32,B: uint32] :
      ( ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),B) != zero_zero_uint32 )
     => ( ( A != zero_zero_uint32 )
        & ( B != zero_zero_uint32 ) ) ) ).

% mult_not_zero
tff(fact_3124_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_3125_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_3126_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_3127_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ) ).

% comm_monoid_mult_class.mult_1
tff(fact_3128_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ) ).

% comm_monoid_mult_class.mult_1
tff(fact_3129_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,one_one_uint32),A) = A ) ).

% comm_monoid_mult_class.mult_1
tff(fact_3130_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ) ).

% comm_monoid_mult_class.mult_1
tff(fact_3131_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ) ).

% comm_monoid_mult_class.mult_1
tff(fact_3132_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ) ).

% comm_monoid_mult_class.mult_1
tff(fact_3133_mult_Ocomm__neutral,axiom,
    ! [A: rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ) ).

% mult.comm_neutral
tff(fact_3134_mult_Ocomm__neutral,axiom,
    ! [A: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ) ).

% mult.comm_neutral
tff(fact_3135_mult_Ocomm__neutral,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),one_one_uint32) = A ) ).

% mult.comm_neutral
tff(fact_3136_mult_Ocomm__neutral,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ) ).

% mult.comm_neutral
tff(fact_3137_mult_Ocomm__neutral,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ) ).

% mult.comm_neutral
tff(fact_3138_mult_Ocomm__neutral,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ) ).

% mult.comm_neutral
tff(fact_3139_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: 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)),Na) )
    <=> ( M = Na ) ) ).

% Suc_mult_cancel1
tff(fact_3140_mult__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Na) = zero_zero_nat ) ).

% mult_0
tff(fact_3141_add__mult__distrib,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),K)) ) ).

% add_mult_distrib
tff(fact_3142_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = 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),Na)) ) ).

% add_mult_distrib2
tff(fact_3143_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_cube
tff(fact_3144_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_3145_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_3146_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_3147_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_3148_nat__mult__1,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Na) = Na ) ).

% nat_mult_1
tff(fact_3149_nat__mult__1__right,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),one_one_nat) = Na ) ).

% nat_mult_1_right
tff(fact_3150_diff__mult__distrib,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),K)) ) ).

% diff_mult_distrib
tff(fact_3151_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = 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),Na)) ) ).

% diff_mult_distrib2
tff(fact_3152_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),A4)) ).

% zero_notin_Suc_image
tff(fact_3153_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3154_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3155_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3156_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3157_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_3158_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_3159_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_3160_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3161_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3162_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3163_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3164_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_3165_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3166_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3167_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_3168_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_3169_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3170_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3171_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_3172_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3173_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3174_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3175_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3176_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_3177_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_3178_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_3179_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_3180_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3181_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3182_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3183_mult__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% mult_right_mono
tff(fact_3184_mult__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% mult_right_mono
tff(fact_3185_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% mult_right_mono
tff(fact_3186_mult__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_right_mono
tff(fact_3187_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_3188_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_3189_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_3190_mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% mult_left_mono
tff(fact_3191_mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% mult_left_mono
tff(fact_3192_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% mult_left_mono
tff(fact_3193_mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_left_mono
tff(fact_3194_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3195_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3196_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3197_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3198_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3199_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3200_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).

% split_mult_pos_le
tff(fact_3201_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).

% split_mult_pos_le
tff(fact_3202_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% split_mult_pos_le
tff(fact_3203_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)) ).

% zero_le_square
tff(fact_3204_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)) ).

% zero_le_square
tff(fact_3205_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_3206_mult__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3207_mult__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3208_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3209_mult__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3210_mult__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3211_mult__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3212_mult__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3213_mult__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3214_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_neg_neg
tff(fact_3215_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_neg_neg
tff(fact_3216_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_3217_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_3218_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_3219_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_3220_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3221_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3222_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3223_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_3224_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_3225_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_3226_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_3227_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_3228_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_3229_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_3230_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_3231_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3232_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3233_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3234_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3235_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_3236_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_3237_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_3238_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_3239_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_3240_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_3241_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_3242_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_3243_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_3244_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_3245_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_3246_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3247_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3248_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3249_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3250_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3251_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3252_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3253_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3254_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3255_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3256_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3257_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3258_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3259_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3260_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3261_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3262_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3263_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3264_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3265_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3266_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3267_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3268_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3269_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_3270_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_3271_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_3272_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_3273_mult__less__cancel__right__disj,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3274_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3275_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3276_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3277_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3278_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3279_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3280_less__1__mult,axiom,
    ! [M: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),Na)) ) ) ).

% less_1_mult
tff(fact_3281_less__1__mult,axiom,
    ! [M: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M),Na)) ) ) ).

% less_1_mult
tff(fact_3282_less__1__mult,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% less_1_mult
tff(fact_3283_less__1__mult,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),Na)) ) ) ).

% less_1_mult
tff(fact_3284_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_3285_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_3286_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_3287_mult__Suc,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ).

% mult_Suc
tff(fact_3288_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_3289_mult__eq__self__implies__10,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
     => ( ( Na = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3290_mlex__snd__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N6: nat] :
      ( ( A = A2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N6)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N6)),B2)) ) ) ).

% mlex_snd_decrI
tff(fact_3291_mlex__fst__decrI,axiom,
    ! [A: nat,A2: nat,B: nat,N6: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),N6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N6)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N6)),B2)) ) ) ) ).

% mlex_fst_decrI
tff(fact_3292_mlex__bound,axiom,
    ! [A: nat,A4: nat,B: nat,N6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),N6)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N6)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N6)) ) ) ).

% mlex_bound
tff(fact_3293_mlex__leI,axiom,
    ! [A: nat,A2: nat,B: nat,B2: nat,N6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N6)),B)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N6)),B2)) ) ) ).

% mlex_leI
tff(fact_3294_field__le__mult__one__interval,axiom,
    ! [Xa: real,Y: real] :
      ( ! [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z3),Xa)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% field_le_mult_one_interval
tff(fact_3295_field__le__mult__one__interval,axiom,
    ! [Xa: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),Xa)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% field_le_mult_one_interval
tff(fact_3296_in__image__insert__iff,axiom,
    ! [B4: set_set_VEBT_VEBT,Xa: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ! [C5: set_VEBT_VEBT] :
          ( member_set_VEBT_VEBT(C5,B4)
         => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),C5) )
     => ( member_set_VEBT_VEBT(A4,image_1661326939266726661T_VEBT(insert_VEBT_VEBT(Xa),B4))
      <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
          & member_set_VEBT_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_3297_in__image__insert__iff,axiom,
    ! [B4: set_set_real,Xa: real,A4: set_real] :
      ( ! [C5: set_real] :
          ( member_set_real(C5,B4)
         => ~ aa_set_real_o(member_real(Xa),C5) )
     => ( member_set_real(A4,image_2436557299294012491t_real(insert_real(Xa),B4))
      <=> ( aa_set_real_o(member_real(Xa),A4)
          & member_set_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_3298_in__image__insert__iff,axiom,
    ! [B4: set_set_set_nat,Xa: set_nat,A4: set_set_nat] :
      ( ! [C5: set_set_nat] :
          ( member_set_set_nat(C5,B4)
         => ~ aa_set_set_nat_o(member_set_nat(Xa),C5) )
     => ( member_set_set_nat(A4,image_7884819252390400639et_nat(insert_set_nat(Xa),B4))
      <=> ( aa_set_set_nat_o(member_set_nat(Xa),A4)
          & member_set_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_3299_in__image__insert__iff,axiom,
    ! [B4: set_set_int,Xa: int,A4: set_int] :
      ( ! [C5: set_int] :
          ( member_set_int(C5,B4)
         => ~ aa_set_int_o(member_int(Xa),C5) )
     => ( member_set_int(A4,image_524474410958335435et_int(insert_int(Xa),B4))
      <=> ( aa_set_int_o(member_int(Xa),A4)
          & member_set_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_3300_in__image__insert__iff,axiom,
    ! [B4: set_set_o,Xa: $o,A4: set_o] :
      ( ! [C5: set_o] :
          ( member_set_o(C5,B4)
         => ~ aa_set_o_o(member_o((Xa)),C5) )
     => ( member_set_o(A4,image_set_o_set_o(insert_o((Xa)),B4))
      <=> ( aa_set_o_o(member_o((Xa)),A4)
          & member_set_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_3301_in__image__insert__iff,axiom,
    ! [B4: set_set_nat,Xa: nat,A4: set_nat] :
      ( ! [C5: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(C5),B4)
         => ~ aa_set_nat_o(member_nat(Xa),C5) )
     => ( aa_set_set_nat_o(member_set_nat(A4),image_7916887816326733075et_nat(insert_nat(Xa),B4))
      <=> ( aa_set_nat_o(member_nat(Xa),A4)
          & aa_set_set_nat_o(member_set_nat(minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_3302_mult__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3303_mult__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3304_mult__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3305_mult__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3306_mult__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3307_mult__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3308_mult__left__less__imp__less,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3309_mult__left__less__imp__less,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3310_mult__left__less__imp__less,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3311_mult__left__less__imp__less,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3312_mult__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3313_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3314_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3315_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3316_mult__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3317_mult__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3318_mult__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3319_mult__right__less__imp__less,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3320_mult__right__less__imp__less,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3321_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3322_mult__right__less__imp__less,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3323_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3324_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3325_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3326_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3327_mult__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3328_mult__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3329_mult__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3330_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3331_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3332_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3333_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3334_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3335_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3336_mult__left__le__imp__le,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3337_mult__left__le__imp__le,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3338_mult__left__le__imp__le,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3339_mult__left__le__imp__le,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3340_mult__right__le__imp__le,axiom,
    ! [A: real,C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3341_mult__right__le__imp__le,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3342_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3343_mult__right__le__imp__le,axiom,
    ! [A: int,C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3344_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3345_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3346_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3347_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3348_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3349_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3350_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3351_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3352_sum__squares__ge__zero,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3353_sum__squares__ge__zero,axiom,
    ! [Xa: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3354_sum__squares__ge__zero,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3355_mult__left__le,axiom,
    ! [C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3356_mult__left__le,axiom,
    ! [C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3357_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3358_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),A) ) ) ).

% mult_left_le
tff(fact_3359_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_3360_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_3361_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_3362_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_3363_mult__right__le__one__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_3364_mult__right__le__one__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Y)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_3365_mult__right__le__one__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Y)),Xa) ) ) ) ).

% mult_right_le_one_le
tff(fact_3366_mult__left__le__one__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_3367_mult__left__le__one__le,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_3368_mult__left__le__one__le,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xa)),Xa) ) ) ) ).

% mult_left_le_one_le
tff(fact_3369_not__sum__squares__lt__zero,axiom,
    ! [Xa: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_3370_not__sum__squares__lt__zero,axiom,
    ! [Xa: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_3371_not__sum__squares__lt__zero,axiom,
    ! [Xa: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_3372_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E2)),C2)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3373_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B)),E2)),C2)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3374_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C2)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3375_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3376_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3377_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3378_less__add__iff2,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3379_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3380_less__add__iff2,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3381_less__add__iff1,axiom,
    ! [A: real,E2: real,C2: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E2)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(A),B)),E2)),C2)),D) ) ).

% less_add_iff1
tff(fact_3382_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C2: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B)),E2)),C2)),D) ) ).

% less_add_iff1
tff(fact_3383_less__add__iff1,axiom,
    ! [A: int,E2: int,C2: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E2)),C2)),D) ) ).

% less_add_iff1
tff(fact_3384_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% one_less_mult
tff(fact_3385_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_3386_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)) ) ) ).

% n_less_n_mult_m
tff(fact_3387_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X_1) ).

% linordered_field_no_ub
tff(fact_3388_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),X_1) ).

% linordered_field_no_ub
tff(fact_3389_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X3) ).

% linordered_field_no_lb
tff(fact_3390_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X3) ).

% linordered_field_no_lb
tff(fact_3391_mult__le__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3392_mult__le__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3393_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3394_mult__le__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3395_mult__le__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3396_mult__le__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3397_mult__le__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3398_mult__le__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3399_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3400_mult__le__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3401_mult__le__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3402_mult__le__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3403_mult__less__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3404_mult__less__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3405_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,C2),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3406_mult__less__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3407_mult__less__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C2),A)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3408_mult__less__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C2),A)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3409_mult__less__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3410_mult__less__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3411_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3412_mult__less__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3413_mult__less__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3414_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3415_convex__bound__le,axiom,
    ! [Xa: real,A: real,Y: real,U: real,V2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V2)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V2) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3416_convex__bound__le,axiom,
    ! [Xa: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V2)
           => ( ( aa_rat_rat(plus_plus_rat(U),V2) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3417_convex__bound__le,axiom,
    ! [Xa: int,A: int,Y: int,U: int,V2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V2)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3418_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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)),Na))) ) ).

% mult_eq_if
tff(fact_3419_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Leaf((A),(B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_3420_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_3421_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)) = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_3422_convex__bound__lt,axiom,
    ! [Xa: real,A: real,Y: real,U: real,V2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V2)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V2) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3423_convex__bound__lt,axiom,
    ! [Xa: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V2)
           => ( ( aa_rat_rat(plus_plus_rat(U),V2) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3424_convex__bound__lt,axiom,
    ! [Xa: int,A: int,Y: int,U: int,V2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V2)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3425_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz)) = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_3426_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Leaf((A),(B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $ite((A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_3427_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2),Uz)) = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_3428_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_3429_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_3430_sum__squares__eq__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_3431_sum__squares__eq__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_3432_sum__squares__eq__zero__iff,axiom,
    ! [Xa: 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,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_3433_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_3434_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_3435_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = 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)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_3436_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = 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)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_3437_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_3438_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_3439_star__false__left,axiom,
    ! [P2: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),P2) = bot_bot_assn ) ).

% star_false_left
tff(fact_3440_star__false__right,axiom,
    ! [P2: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),bot_bot_assn) = bot_bot_assn ) ).

% star_false_right
tff(fact_3441_snga__same__false,axiom,
    ! [P3: array_VEBT_VEBTi,Xa: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(P3,Xa)),snga_assn_VEBT_VEBTi(P3,Y)) = bot_bot_assn ) ).

% snga_same_false
tff(fact_3442_mod__h__bot__iff_I5_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Q2)),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P2),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(rep_assn(Q2),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(5)
tff(fact_3443_mod__starE,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),H2)
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(rep_assn(A),X_1)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(B),H_2) ) ) ).

% mod_starE
tff(fact_3444_mod__starD,axiom,
    ! [A4: assn,B4: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),B4)),H2)
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(A4),H1)
          & aa_Pro653082453696054616_nat_o(rep_assn(B4),H22) ) ) ).

% mod_starD
tff(fact_3445_assn__one__left,axiom,
    ! [P2: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),P2) = P2 ) ).

% assn_one_left
tff(fact_3446_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: 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),Na) )
    <=> ( ( K = zero_zero_nat )
        | ( M = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_3447_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_3448_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
      <=> ( M = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_3449_sum__squares__le__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)
    <=> ( ( Xa = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3450_sum__squares__le__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)
    <=> ( ( Xa = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3451_sum__squares__le__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( Xa = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3452_sum__squares__gt__zero__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))
    <=> ( ( Xa != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3453_sum__squares__gt__zero__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))
    <=> ( ( Xa != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3454_sum__squares__gt__zero__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( Xa != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3455_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_3456_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( 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) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_3457_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( 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)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_3458_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3459_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Xa)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3460_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Xa)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3461_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3462_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3463_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3464_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C2: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),C2)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3465_add__scale__eq__noteq,axiom,
    ! [R: real,A: real,B: real,C2: real,D: real] :
      ( ( R != zero_zero_real )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),C2)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3466_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C2: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),C2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3467_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C2: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R),C2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3468_mult__less__iff1,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_3469_mult__less__iff1,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_3470_mult__less__iff1,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Y) ) ) ).

% mult_less_iff1
tff(fact_3471_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_3472_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_3473_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_3474_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_3475_mul__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xa)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_3476_minus__Max__eq__Min,axiom,
    ! [S2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,lattic4901227151466704046nteger(S2)) = lattic1063845414844153500nteger(image_4470545334726330049nteger(uminus1351360451143612070nteger,S2)) ) ) ) ).

% minus_Max_eq_Min
tff(fact_3477_minus__Max__eq__Min,axiom,
    ! [S2: set_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( ( S2 != bot_bot_set_real )
       => ( aa_real_real(uminus_uminus_real,lattic4275903605611617917x_real(S2)) = lattic3629708407755379051n_real(image_real_real(uminus_uminus_real,S2)) ) ) ) ).

% minus_Max_eq_Min
tff(fact_3478_minus__Max__eq__Min,axiom,
    ! [S2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S2)
     => ( ( S2 != bot_bot_set_rat )
       => ( aa_rat_rat(uminus_uminus_rat,lattic7630753665789217321ax_rat(S2)) = lattic8086005427650270231in_rat(image_rat_rat(uminus_uminus_rat,S2)) ) ) ) ).

% minus_Max_eq_Min
tff(fact_3479_minus__Max__eq__Min,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ( aa_int_int(uminus_uminus_int,lattic8263393255366662781ax_int(S2)) = lattic8718645017227715691in_int(aa_set_int_set_int(image_int_int(uminus_uminus_int),S2)) ) ) ) ).

% minus_Max_eq_Min
tff(fact_3480_minus__Min__eq__Max,axiom,
    ! [S2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,lattic1063845414844153500nteger(S2)) = lattic4901227151466704046nteger(image_4470545334726330049nteger(uminus1351360451143612070nteger,S2)) ) ) ) ).

% minus_Min_eq_Max
tff(fact_3481_minus__Min__eq__Max,axiom,
    ! [S2: set_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( ( S2 != bot_bot_set_real )
       => ( aa_real_real(uminus_uminus_real,lattic3629708407755379051n_real(S2)) = lattic4275903605611617917x_real(image_real_real(uminus_uminus_real,S2)) ) ) ) ).

% minus_Min_eq_Max
tff(fact_3482_minus__Min__eq__Max,axiom,
    ! [S2: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S2)
     => ( ( S2 != bot_bot_set_rat )
       => ( aa_rat_rat(uminus_uminus_rat,lattic8086005427650270231in_rat(S2)) = lattic7630753665789217321ax_rat(image_rat_rat(uminus_uminus_rat,S2)) ) ) ) ).

% minus_Min_eq_Max
tff(fact_3483_minus__Min__eq__Max,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ( aa_int_int(uminus_uminus_int,lattic8718645017227715691in_int(S2)) = lattic8263393255366662781ax_int(aa_set_int_set_int(image_int_int(uminus_uminus_int),S2)) ) ) ) ).

% minus_Min_eq_Max
tff(fact_3484_add__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xa)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_3485_add_Oinverse__inverse,axiom,
    ! [A: uint32] : ( uminus_uminus_uint32(uminus_uminus_uint32(A)) = A ) ).

% add.inverse_inverse
tff(fact_3486_add_Oinverse__inverse,axiom,
    ! [A: real] : ( aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A)) = A ) ).

% add.inverse_inverse
tff(fact_3487_add_Oinverse__inverse,axiom,
    ! [A: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ) ).

% add.inverse_inverse
tff(fact_3488_add_Oinverse__inverse,axiom,
    ! [A: int] : ( aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ) ).

% add.inverse_inverse
tff(fact_3489_neg__equal__iff__equal,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32(A) = uminus_uminus_uint32(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3490_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3491_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3492_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_3493_Compl__subset__Compl__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(A4)),uminus1532241313380277803et_int(B4))
    <=> aa_set_int_o(ord_less_eq_set_int(B4),A4) ) ).

% Compl_subset_Compl_iff
tff(fact_3494_Compl__anti__mono,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(B4)),uminus1532241313380277803et_int(A4)) ) ).

% Compl_anti_mono
tff(fact_3495_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_3496_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_3497_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_3498_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_3499_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_3500_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_3501_neg__equal__0__iff__equal,axiom,
    ! [A: uint32] :
      ( ( uminus_uminus_uint32(A) = zero_zero_uint32 )
    <=> ( A = zero_zero_uint32 ) ) ).

% neg_equal_0_iff_equal
tff(fact_3502_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_3503_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_3504_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_3505_neg__0__equal__iff__equal,axiom,
    ! [A: uint32] :
      ( ( zero_zero_uint32 = uminus_uminus_uint32(A) )
    <=> ( zero_zero_uint32 = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3506_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A) )
    <=> ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3507_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3508_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_3509_add_Oinverse__neutral,axiom,
    uminus_uminus_uint32(zero_zero_uint32) = zero_zero_uint32 ).

% add.inverse_neutral
tff(fact_3510_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_3511_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_3512_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_3513_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_3514_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_3515_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_3516_compl__le__compl__iff,axiom,
    ! [Xa: assn,Y: assn] :
      ( ord_less_eq_assn(uminus_uminus_assn(Xa),uminus_uminus_assn(Y))
    <=> ord_less_eq_assn(Y,Xa) ) ).

% compl_le_compl_iff
tff(fact_3517_compl__le__compl__iff,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xa)),uminus1532241313380277803et_int(Y))
    <=> aa_set_int_o(ord_less_eq_set_int(Y),Xa) ) ).

% compl_le_compl_iff
tff(fact_3518_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_3519_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_3520_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_3521_compl__less__compl__iff,axiom,
    ! [Xa: assn,Y: assn] :
      ( ord_less_assn(uminus_uminus_assn(Xa),uminus_uminus_assn(Y))
    <=> ord_less_assn(Y,Xa) ) ).

% compl_less_compl_iff
tff(fact_3522_add__minus__cancel,axiom,
    ! [A: uint32,B: uint32] : ( aa_uint32_uint32(plus_plus_uint32(A),aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(A)),B)) = B ) ).

% add_minus_cancel
tff(fact_3523_add__minus__cancel,axiom,
    ! [A: real,B: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ) ).

% add_minus_cancel
tff(fact_3524_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ) ).

% add_minus_cancel
tff(fact_3525_add__minus__cancel,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ) ).

% add_minus_cancel
tff(fact_3526_minus__add__cancel,axiom,
    ! [A: uint32,B: uint32] : ( aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(A)),aa_uint32_uint32(plus_plus_uint32(A),B)) = B ) ).

% minus_add_cancel
tff(fact_3527_minus__add__cancel,axiom,
    ! [A: real,B: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ) ).

% minus_add_cancel
tff(fact_3528_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ) ).

% minus_add_cancel
tff(fact_3529_minus__add__cancel,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ) ).

% minus_add_cancel
tff(fact_3530_minus__add__distrib,axiom,
    ! [A: uint32,B: uint32] : ( uminus_uminus_uint32(aa_uint32_uint32(plus_plus_uint32(A),B)) = aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(A)),uminus_uminus_uint32(B)) ) ).

% minus_add_distrib
tff(fact_3531_minus__add__distrib,axiom,
    ! [A: real,B: real] : ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ) ).

% minus_add_distrib
tff(fact_3532_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ) ).

% minus_add_distrib
tff(fact_3533_minus__add__distrib,axiom,
    ! [A: int,B: int] : ( aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ) ).

% minus_add_distrib
tff(fact_3534_minus__diff__eq,axiom,
    ! [A: uint32,B: uint32] : ( uminus_uminus_uint32(minus_minus_uint32(A,B)) = minus_minus_uint32(B,A) ) ).

% minus_diff_eq
tff(fact_3535_minus__diff__eq,axiom,
    ! [A: real,B: real] : ( aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(minus_minus_real(B),A) ) ).

% minus_diff_eq
tff(fact_3536_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ) ).

% minus_diff_eq
tff(fact_3537_minus__diff__eq,axiom,
    ! [A: int,B: int] : ( aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ) ).

% minus_diff_eq
tff(fact_3538_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_3539_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_3540_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_3541_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_le_0_iff_le
tff(fact_3542_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_3543_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_3544_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_3545_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_3546_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_3547_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_less_eq_nonneg
tff(fact_3548_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_3549_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_3550_less__neg__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% less_neg_neg
tff(fact_3551_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_3552_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_3553_neg__less__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_pos
tff(fact_3554_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_3555_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_3556_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_3557_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_3558_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_3559_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_0_iff_less
tff(fact_3560_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_3561_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_3562_ab__left__minus,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(A)),A) = zero_zero_uint32 ) ).

% ab_left_minus
tff(fact_3563_ab__left__minus,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ) ).

% ab_left_minus
tff(fact_3564_ab__left__minus,axiom,
    ! [A: rat] : ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ) ).

% ab_left_minus
tff(fact_3565_ab__left__minus,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ) ).

% ab_left_minus
tff(fact_3566_add_Oright__inverse,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(plus_plus_uint32(A),uminus_uminus_uint32(A)) = zero_zero_uint32 ) ).

% add.right_inverse
tff(fact_3567_add_Oright__inverse,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ) ).

% add.right_inverse
tff(fact_3568_add_Oright__inverse,axiom,
    ! [A: rat] : ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ) ).

% add.right_inverse
tff(fact_3569_add_Oright__inverse,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ) ).

% add.right_inverse
tff(fact_3570_diff__0,axiom,
    ! [A: uint32] : ( minus_minus_uint32(zero_zero_uint32,A) = uminus_uminus_uint32(A) ) ).

% diff_0
tff(fact_3571_diff__0,axiom,
    ! [A: real] : ( aa_real_real(minus_minus_real(zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ) ).

% diff_0
tff(fact_3572_diff__0,axiom,
    ! [A: rat] : ( aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ) ).

% diff_0
tff(fact_3573_diff__0,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ) ).

% diff_0
tff(fact_3574_verit__minus__simplify_I3_J,axiom,
    ! [B: uint32] : ( minus_minus_uint32(zero_zero_uint32,B) = uminus_uminus_uint32(B) ) ).

% verit_minus_simplify(3)
tff(fact_3575_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : ( aa_real_real(minus_minus_real(zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ) ).

% verit_minus_simplify(3)
tff(fact_3576_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : ( aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ) ).

% verit_minus_simplify(3)
tff(fact_3577_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ) ).

% verit_minus_simplify(3)
tff(fact_3578_uminus__add__conv__diff,axiom,
    ! [A: uint32,B: uint32] : ( aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(A)),B) = minus_minus_uint32(B,A) ) ).

% uminus_add_conv_diff
tff(fact_3579_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(minus_minus_real(B),A) ) ).

% uminus_add_conv_diff
tff(fact_3580_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ) ).

% uminus_add_conv_diff
tff(fact_3581_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ) ).

% uminus_add_conv_diff
tff(fact_3582_diff__minus__eq__add,axiom,
    ! [A: uint32,B: uint32] : ( minus_minus_uint32(A,uminus_uminus_uint32(B)) = aa_uint32_uint32(plus_plus_uint32(A),B) ) ).

% diff_minus_eq_add
tff(fact_3583_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : ( aa_real_real(minus_minus_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ) ).

% diff_minus_eq_add
tff(fact_3584_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ) ).

% diff_minus_eq_add
tff(fact_3585_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ) ).

% diff_minus_eq_add
tff(fact_3586_subset__Compl__singleton,axiom,
    ! [A4: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B),bot_bo8194388402131092736T_VEBT)))
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),A4) ) ).

% subset_Compl_singleton
tff(fact_3587_subset__Compl__singleton,axiom,
    ! [A4: set_real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),uminus612125837232591019t_real(aa_set_real_set_real(insert_real(B),bot_bot_set_real)))
    <=> ~ aa_set_real_o(member_real(B),A4) ) ).

% subset_Compl_singleton
tff(fact_3588_subset__Compl__singleton,axiom,
    ! [A4: set_set_nat,B: set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),uminus613421341184616069et_nat(aa_set1013257548857649334et_nat(insert_set_nat(B),bot_bot_set_set_nat)))
    <=> ~ aa_set_set_nat_o(member_set_nat(B),A4) ) ).

% subset_Compl_singleton
tff(fact_3589_subset__Compl__singleton,axiom,
    ! [A4: set_nat,B: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),uminus5710092332889474511et_nat(aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B),A4) ) ).

% subset_Compl_singleton
tff(fact_3590_subset__Compl__singleton,axiom,
    ! [A4: set_o,B: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),uminus_uminus_set_o(aa_set_o_set_o(insert_o((B)),bot_bot_set_o)))
    <=> ~ aa_set_o_o(member_o((B)),A4) ) ).

% subset_Compl_singleton
tff(fact_3591_subset__Compl__singleton,axiom,
    ! [A4: set_int,B: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(B),bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B),A4) ) ).

% subset_Compl_singleton
tff(fact_3592_add__neg__numeral__special_I7_J,axiom,
    aa_uint32_uint32(plus_plus_uint32(one_one_uint32),uminus_uminus_uint32(one_one_uint32)) = zero_zero_uint32 ).

% add_neg_numeral_special(7)
tff(fact_3593_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_3594_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_3595_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_3596_add__neg__numeral__special_I8_J,axiom,
    aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(one_one_uint32)),one_one_uint32) = zero_zero_uint32 ).

% add_neg_numeral_special(8)
tff(fact_3597_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_3598_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_3599_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_3600_diff__numeral__special_I12_J,axiom,
    minus_minus_uint32(uminus_uminus_uint32(one_one_uint32),uminus_uminus_uint32(one_one_uint32)) = zero_zero_uint32 ).

% diff_numeral_special(12)
tff(fact_3601_diff__numeral__special_I12_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_3602_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_3603_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_3604_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_3605_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_3606_assn__times__comm,axiom,
    ! [P2: assn,Q2: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Q2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q2),P2) ) ).

% assn_times_comm
tff(fact_3607_assn__times__assoc,axiom,
    ! [P2: assn,Q2: assn,R2: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Q2)),R2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q2),R2)) ) ).

% assn_times_assoc
tff(fact_3608_equation__minus__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A = uminus_uminus_uint32(B) )
    <=> ( B = uminus_uminus_uint32(A) ) ) ).

% equation_minus_iff
tff(fact_3609_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_3610_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_3611_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_3612_minus__equation__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32(A) = B )
    <=> ( uminus_uminus_uint32(B) = A ) ) ).

% minus_equation_iff
tff(fact_3613_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(uminus_uminus_real,B) = A ) ) ).

% minus_equation_iff
tff(fact_3614_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_3615_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_3616_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_3617_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_3618_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_3619_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_3620_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_le_iff
tff(fact_3621_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_3622_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_3623_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_3624_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_3625_compl__le__swap2,axiom,
    ! [Y: assn,Xa: assn] :
      ( ord_less_eq_assn(uminus_uminus_assn(Y),Xa)
     => ord_less_eq_assn(uminus_uminus_assn(Xa),Y) ) ).

% compl_le_swap2
tff(fact_3626_compl__le__swap2,axiom,
    ! [Y: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),Xa)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xa)),Y) ) ).

% compl_le_swap2
tff(fact_3627_compl__le__swap1,axiom,
    ! [Y: assn,Xa: assn] :
      ( ord_less_eq_assn(Y,uminus_uminus_assn(Xa))
     => ord_less_eq_assn(Xa,uminus_uminus_assn(Y)) ) ).

% compl_le_swap1
tff(fact_3628_compl__le__swap1,axiom,
    ! [Y: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),uminus1532241313380277803et_int(Xa))
     => aa_set_int_o(ord_less_eq_set_int(Xa),uminus1532241313380277803et_int(Y)) ) ).

% compl_le_swap1
tff(fact_3629_compl__mono,axiom,
    ! [Xa: assn,Y: assn] :
      ( ord_less_eq_assn(Xa,Y)
     => ord_less_eq_assn(uminus_uminus_assn(Y),uminus_uminus_assn(Xa)) ) ).

% compl_mono
tff(fact_3630_compl__mono,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(Xa)) ) ).

% compl_mono
tff(fact_3631_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_3632_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_less_iff
tff(fact_3633_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_3634_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_3635_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_3636_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_3637_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_3638_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_3639_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_3640_compl__less__swap2,axiom,
    ! [Y: assn,Xa: assn] :
      ( ord_less_assn(uminus_uminus_assn(Y),Xa)
     => ord_less_assn(uminus_uminus_assn(Xa),Y) ) ).

% compl_less_swap2
tff(fact_3641_compl__less__swap1,axiom,
    ! [Y: assn,Xa: assn] :
      ( ord_less_assn(Y,uminus_uminus_assn(Xa))
     => ord_less_assn(Xa,uminus_uminus_assn(Y)) ) ).

% compl_less_swap1
tff(fact_3642_group__cancel_Oneg1,axiom,
    ! [A4: uint32,K: uint32,A: uint32] :
      ( ( A4 = aa_uint32_uint32(plus_plus_uint32(K),A) )
     => ( uminus_uminus_uint32(A4) = aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(K)),uminus_uminus_uint32(A)) ) ) ).

% group_cancel.neg1
tff(fact_3643_group__cancel_Oneg1,axiom,
    ! [A4: real,K: real,A: real] :
      ( ( A4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_3644_group__cancel_Oneg1,axiom,
    ! [A4: rat,K: rat,A: rat] :
      ( ( A4 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A4) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_3645_group__cancel_Oneg1,axiom,
    ! [A4: int,K: int,A: int] :
      ( ( A4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_3646_add_Oinverse__distrib__swap,axiom,
    ! [A: uint32,B: uint32] : ( uminus_uminus_uint32(aa_uint32_uint32(plus_plus_uint32(A),B)) = aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(B)),uminus_uminus_uint32(A)) ) ).

% add.inverse_distrib_swap
tff(fact_3647_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% add.inverse_distrib_swap
tff(fact_3648_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% add.inverse_distrib_swap
tff(fact_3649_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : ( aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% add.inverse_distrib_swap
tff(fact_3650_minus__diff__commute,axiom,
    ! [B: uint32,A: uint32] : ( minus_minus_uint32(uminus_uminus_uint32(B),A) = minus_minus_uint32(uminus_uminus_uint32(A),B) ) ).

% minus_diff_commute
tff(fact_3651_minus__diff__commute,axiom,
    ! [B: real,A: real] : ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),B) ) ).

% minus_diff_commute
tff(fact_3652_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% minus_diff_commute
tff(fact_3653_minus__diff__commute,axiom,
    ! [B: int,A: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% minus_diff_commute
tff(fact_3654_neg__eq__iff__add__eq__0,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32(A) = B )
    <=> ( aa_uint32_uint32(plus_plus_uint32(A),B) = zero_zero_uint32 ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3655_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3656_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3657_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3658_eq__neg__iff__add__eq__0,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A = uminus_uminus_uint32(B) )
    <=> ( aa_uint32_uint32(plus_plus_uint32(A),B) = zero_zero_uint32 ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3659_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3660_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3661_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3662_add_Oinverse__unique,axiom,
    ! [A: uint32,B: uint32] :
      ( ( aa_uint32_uint32(plus_plus_uint32(A),B) = zero_zero_uint32 )
     => ( uminus_uminus_uint32(A) = B ) ) ).

% add.inverse_unique
tff(fact_3663_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_3664_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_3665_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_3666_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: uint32] : ( aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(A)),A) = zero_zero_uint32 ) ).

% ab_group_add_class.ab_left_minus
tff(fact_3667_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
tff(fact_3668_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
tff(fact_3669_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
tff(fact_3670_add__eq__0__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( aa_uint32_uint32(plus_plus_uint32(A),B) = zero_zero_uint32 )
    <=> ( B = uminus_uminus_uint32(A) ) ) ).

% add_eq_0_iff
tff(fact_3671_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_3672_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_3673_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_3674_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_3675_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_3676_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_3677_le__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% le_minus_one_simps(2)
tff(fact_3678_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_3679_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_3680_le__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(4)
tff(fact_3681_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_3682_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_3683_less__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(4)
tff(fact_3684_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_3685_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_3686_less__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% less_minus_one_simps(2)
tff(fact_3687_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_3688_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_3689_group__cancel_Osub2,axiom,
    ! [B4: uint32,K: uint32,B: uint32,A: uint32] :
      ( ( B4 = aa_uint32_uint32(plus_plus_uint32(K),B) )
     => ( minus_minus_uint32(A,B4) = aa_uint32_uint32(plus_plus_uint32(uminus_uminus_uint32(K)),minus_minus_uint32(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_3690_group__cancel_Osub2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(minus_minus_real(A),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_3691_group__cancel_Osub2,axiom,
    ! [B4: rat,K: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B4) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_3692_group__cancel_Osub2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_3693_diff__conv__add__uminus,axiom,
    ! [A: uint32,B: uint32] : ( minus_minus_uint32(A,B) = aa_uint32_uint32(plus_plus_uint32(A),uminus_uminus_uint32(B)) ) ).

% diff_conv_add_uminus
tff(fact_3694_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : ( aa_real_real(minus_minus_real(A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ) ).

% diff_conv_add_uminus
tff(fact_3695_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ).

% diff_conv_add_uminus
tff(fact_3696_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ) ).

% diff_conv_add_uminus
tff(fact_3697_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: uint32,B: uint32] : ( minus_minus_uint32(A,B) = aa_uint32_uint32(plus_plus_uint32(A),uminus_uminus_uint32(B)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3698_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : ( aa_real_real(minus_minus_real(A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3699_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3700_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3701_subset__Compl__self__eq,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),uminus5710092332889474511et_nat(A4))
    <=> ( A4 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_3702_subset__Compl__self__eq,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),uminus_uminus_set_o(A4))
    <=> ( A4 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_3703_subset__Compl__self__eq,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),uminus1532241313380277803et_int(A4))
    <=> ( A4 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_3704_le__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% le_minus_one_simps(1)
tff(fact_3705_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_3706_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_3707_le__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(3)
tff(fact_3708_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_3709_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_3710_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_3711_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_3712_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_3713_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_3714_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_3715_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_3716_Compl__insert,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT] : ( uminus8041839845116263051T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = minus_5127226145743854075T_VEBT(uminus8041839845116263051T_VEBT(A4),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)) ) ).

% Compl_insert
tff(fact_3717_Compl__insert,axiom,
    ! [Xa: int,A4: set_int] : ( uminus1532241313380277803et_int(aa_set_int_set_int(insert_int(Xa),A4)) = minus_minus_set_int(uminus1532241313380277803et_int(A4),aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) ) ).

% Compl_insert
tff(fact_3718_Compl__insert,axiom,
    ! [Xa: $o,A4: set_o] : ( uminus_uminus_set_o(aa_set_o_set_o(insert_o((Xa)),A4)) = minus_minus_set_o(uminus_uminus_set_o(A4),aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)) ) ).

% Compl_insert
tff(fact_3719_Compl__insert,axiom,
    ! [Xa: nat,A4: set_nat] : ( uminus5710092332889474511et_nat(aa_set_nat_set_nat(insert_nat(Xa),A4)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A4),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) ) ).

% Compl_insert
tff(fact_3720_frac__neg,axiom,
    ! [Xa: real] :
      ( aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,Xa)) = $ite(aa_set_real_o(member_real(Xa),ring_1_Ints_real),zero_zero_real,aa_real_real(minus_minus_real(one_one_real),aa_real_real(archim2898591450579166408c_real,Xa))) ) ).

% frac_neg
tff(fact_3721_frac__neg,axiom,
    ! [Xa: rat] :
      ( aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,Xa)) = $ite(aa_set_rat_o(member_rat(Xa),ring_1_Ints_rat),zero_zero_rat,aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(archimedean_frac_rat,Xa))) ) ).

% frac_neg
tff(fact_3722_dbl__dec__simps_I2_J,axiom,
    neg_nu965353292909893953uint32(zero_zero_uint32) = uminus_uminus_uint32(one_one_uint32) ).

% dbl_dec_simps(2)
tff(fact_3723_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_3724_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_3725_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_3726_foldr__one,axiom,
    ! [D: nat,Ys: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ).

% foldr_one
tff(fact_3727_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M7)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M7,aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_3728_Gcd__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( aa_set_nat_nat(gcd_Gcd_nat,A4) = zero_zero_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ) ).

% Gcd_0_iff
tff(fact_3729_Gcd__0__iff,axiom,
    ! [A4: set_int] :
      ( ( gcd_Gcd_int(A4) = zero_zero_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(insert_int(zero_zero_int),bot_bot_set_int)) ) ).

% Gcd_0_iff
tff(fact_3730_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,vEBT_Leaf((A),(B))),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $ite(Xa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_3731_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,Sa: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),Sa)),Xa) = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_3732_the__dflt__None__empty,axiom,
    dflt_None_set_nat(bot_bot_set_nat) = none_set_nat ).

% the_dflt_None_empty
tff(fact_3733_the__dflt__None__empty,axiom,
    dflt_None_set_int(bot_bot_set_int) = none_set_int ).

% the_dflt_None_empty
tff(fact_3734_the__dflt__None__empty,axiom,
    dflt_None_set_o(bot_bot_set_o) = none_set_o ).

% the_dflt_None_empty
tff(fact_3735_ComplI,axiom,
    ! [C2: nat,A4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(C2),A4)
     => aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A4)) ) ).

% ComplI
tff(fact_3736_ComplI,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A4)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),uminus8041839845116263051T_VEBT(A4)) ) ).

% ComplI
tff(fact_3737_ComplI,axiom,
    ! [C2: real,A4: set_real] :
      ( ~ aa_set_real_o(member_real(C2),A4)
     => aa_set_real_o(member_real(C2),uminus612125837232591019t_real(A4)) ) ).

% ComplI
tff(fact_3738_ComplI,axiom,
    ! [C2: int,A4: set_int] :
      ( ~ aa_set_int_o(member_int(C2),A4)
     => aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A4)) ) ).

% ComplI
tff(fact_3739_ComplI,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(C2),A4)
     => aa_set_set_nat_o(member_set_nat(C2),uminus613421341184616069et_nat(A4)) ) ).

% ComplI
tff(fact_3740_Compl__iff,axiom,
    ! [C2: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A4))
    <=> ~ aa_set_nat_o(member_nat(C2),A4) ) ).

% Compl_iff
tff(fact_3741_Compl__iff,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),uminus8041839845116263051T_VEBT(A4))
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A4) ) ).

% Compl_iff
tff(fact_3742_Compl__iff,axiom,
    ! [C2: real,A4: set_real] :
      ( aa_set_real_o(member_real(C2),uminus612125837232591019t_real(A4))
    <=> ~ aa_set_real_o(member_real(C2),A4) ) ).

% Compl_iff
tff(fact_3743_Compl__iff,axiom,
    ! [C2: int,A4: set_int] :
      ( aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A4))
    <=> ~ aa_set_int_o(member_int(C2),A4) ) ).

% Compl_iff
tff(fact_3744_Compl__iff,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),uminus613421341184616069et_nat(A4))
    <=> ~ aa_set_set_nat_o(member_set_nat(C2),A4) ) ).

% Compl_iff
tff(fact_3745_the__dflt__None__nonempty,axiom,
    ! [S2: set_nat] :
      ( ( S2 != bot_bot_set_nat )
     => ( dflt_None_set_nat(S2) = some_set_nat(S2) ) ) ).

% the_dflt_None_nonempty
tff(fact_3746_the__dflt__None__nonempty,axiom,
    ! [S2: set_int] :
      ( ( S2 != bot_bot_set_int )
     => ( dflt_None_set_int(S2) = some_set_int(S2) ) ) ).

% the_dflt_None_nonempty
tff(fact_3747_the__dflt__None__nonempty,axiom,
    ! [S2: set_o] :
      ( ( S2 != bot_bot_set_o )
     => ( dflt_None_set_o(S2) = some_set_o(S2) ) ) ).

% the_dflt_None_nonempty
tff(fact_3748_Gcd__empty,axiom,
    aa_set_nat_nat(gcd_Gcd_nat,bot_bot_set_nat) = zero_zero_nat ).

% Gcd_empty
tff(fact_3749_Gcd__empty,axiom,
    gcd_Gcd_int(bot_bot_set_int) = zero_zero_int ).

% Gcd_empty
tff(fact_3750_ComplD,axiom,
    ! [C2: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A4))
     => ~ aa_set_nat_o(member_nat(C2),A4) ) ).

% ComplD
tff(fact_3751_ComplD,axiom,
    ! [C2: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),uminus8041839845116263051T_VEBT(A4))
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A4) ) ).

% ComplD
tff(fact_3752_ComplD,axiom,
    ! [C2: real,A4: set_real] :
      ( aa_set_real_o(member_real(C2),uminus612125837232591019t_real(A4))
     => ~ aa_set_real_o(member_real(C2),A4) ) ).

% ComplD
tff(fact_3753_ComplD,axiom,
    ! [C2: int,A4: set_int] :
      ( aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A4))
     => ~ aa_set_int_o(member_int(C2),A4) ) ).

% ComplD
tff(fact_3754_ComplD,axiom,
    ! [C2: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(C2),uminus613421341184616069et_nat(A4))
     => ~ aa_set_set_nat_o(member_set_nat(C2),A4) ) ).

% ComplD
tff(fact_3755_dflt__None__set__def,axiom,
    ! [S2: set_nat] :
      ( dflt_None_set_nat(S2) = $ite(S2 = bot_bot_set_nat,none_set_nat,some_set_nat(S2)) ) ).

% dflt_None_set_def
tff(fact_3756_dflt__None__set__def,axiom,
    ! [S2: set_int] :
      ( dflt_None_set_int(S2) = $ite(S2 = bot_bot_set_int,none_set_int,some_set_int(S2)) ) ).

% dflt_None_set_def
tff(fact_3757_dflt__None__set__def,axiom,
    ! [S2: set_o] :
      ( dflt_None_set_o(S2) = $ite(S2 = bot_bot_set_o,none_set_o,some_set_o(S2)) ) ).

% dflt_None_set_def
tff(fact_3758_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,Sa: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),Sa)),Xa) = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_3759_the__dflt__None__set,axiom,
    ! [Xa: set_nat] : ( the_default_set_nat(bot_bot_set_nat,dflt_None_set_nat(Xa)) = Xa ) ).

% the_dflt_None_set
tff(fact_3760_the__dflt__None__set,axiom,
    ! [Xa: set_int] : ( the_default_set_int(bot_bot_set_int,dflt_None_set_int(Xa)) = Xa ) ).

% the_dflt_None_set
tff(fact_3761_the__dflt__None__set,axiom,
    ! [Xa: set_o] : ( the_default_set_o(bot_bot_set_o,dflt_None_set_o(Xa)) = Xa ) ).

% the_dflt_None_set
tff(fact_3762_foldr__same__int,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ! [X: nat,Y3: nat] :
          ( aa_set_nat_o(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs))
         => ( aa_set_nat_o(member_nat(Y3),aa_list_nat_set_nat(set_nat2,Xs))
           => ( X = Y3 ) ) )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs))
           => ( X = Y ) )
       => ( aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_list_nat_nat(size_size_list_nat,Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_3763_Gcd__fin__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( aa_set_nat_nat(semiri4258706085729940814in_nat,A4) = zero_zero_nat )
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat))
        & aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% Gcd_fin_0_iff
tff(fact_3764_Gcd__fin__0__iff,axiom,
    ! [A4: set_int] :
      ( ( semiri4256215615220890538in_int(A4) = zero_zero_int )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(insert_int(zero_zero_int),bot_bot_set_int))
        & aa_set_int_o(finite_finite_int,A4) ) ) ).

% Gcd_fin_0_iff
tff(fact_3765_power__decreasing__iff,axiom,
    ! [B: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3766_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3767_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3768_power__decreasing__iff,axiom,
    ! [B: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3769_card__insert__le__m1,axiom,
    ! [Na: nat,Y: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3770_card__insert__le__m1,axiom,
    ! [Na: nat,Y: set_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(insert_int(Xa),Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3771_card__insert__le__m1,axiom,
    ! [Na: nat,Y: set_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(aa_set_o_set_o(insert_o((Xa)),Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3772_card__insert__le__m1,axiom,
    ! [Na: nat,Y: set_complex,Xa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(insert_complex(Xa,Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3773_card__insert__le__m1,axiom,
    ! [Na: nat,Y: set_literal,Xa: literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(insert_literal(Xa,Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3774_card__insert__le__m1,axiom,
    ! [Na: nat,Y: set_list_nat,Xa: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(insert_list_nat(Xa,Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3775_card__insert__le__m1,axiom,
    ! [Na: nat,Y: set_set_nat,Xa: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3776_card__insert__le__m1,axiom,
    ! [Na: nat,Y: set_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,Y)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(insert_nat(Xa),Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_3777_count__notin,axiom,
    ! [Xa: int,Xs: list_int] :
      ( ~ aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
     => ( aa_int_nat(count_list_int(Xs),Xa) = zero_zero_nat ) ) ).

% count_notin
tff(fact_3778_count__notin,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),Xa) = zero_zero_nat ) ) ).

% count_notin
tff(fact_3779_count__notin,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),Xa) = zero_zero_nat ) ) ).

% count_notin
tff(fact_3780_count__notin,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs))
     => ( aa_nat_nat(count_list_nat(Xs),Xa) = zero_zero_nat ) ) ).

% count_notin
tff(fact_3781_count__notin,axiom,
    ! [Xa: real,Xs: list_real] :
      ( ~ aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,Xs))
     => ( aa_real_nat(count_list_real(Xs),Xa) = zero_zero_nat ) ) ).

% count_notin
tff(fact_3782_count__notin,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( ~ aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,Xs))
     => ( aa_o_nat(count_list_o(Xs),(Xa)) = zero_zero_nat ) ) ).

% count_notin
tff(fact_3783_image__Fpow__mono,axiom,
    ! [F: fun_nat_set_nat,A4: set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(F),A4)),B4)
     => ord_le9131159989063066194et_nat(image_6725021117256019401et_nat(image_nat_set_nat(F),finite_Fpow_nat(A4)),finite_Fpow_set_nat(B4)) ) ).

% image_Fpow_mono
tff(fact_3784_image__Fpow__mono,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)),B4)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(image_7916887816326733075et_nat(image_nat_nat(F),finite_Fpow_nat(A4))),finite_Fpow_nat(B4)) ) ).

% image_Fpow_mono
tff(fact_3785_image__Fpow__mono,axiom,
    ! [F: fun_int_nat,A4: set_int,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_int_set_nat(image_int_nat(F),A4)),B4)
     => aa_set_set_nat_o(ord_le6893508408891458716et_nat(image_4702325430467532143et_nat(image_int_nat(F),finite_Fpow_int(A4))),finite_Fpow_nat(B4)) ) ).

% image_Fpow_mono
tff(fact_3786_image__Fpow__mono,axiom,
    ! [F: fun_nat_int,A4: set_nat,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B4)
     => ord_le4403425263959731960et_int(image_3739036796817536367et_int(image_nat_int(F),finite_Fpow_nat(A4)),finite_Fpow_int(B4)) ) ).

% image_Fpow_mono
tff(fact_3787_image__Fpow__mono,axiom,
    ! [F: fun_int_int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B4)
     => ord_le4403425263959731960et_int(image_524474410958335435et_int(image_int_int(F),finite_Fpow_int(A4)),finite_Fpow_int(B4)) ) ).

% image_Fpow_mono
tff(fact_3788_power__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xa)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_3789_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_3790_power__Suc__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(suc,zero_zero_nat) ) ).

% power_Suc_0
tff(fact_3791_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( Xa = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_3792_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xa),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_3793_power__inject__exp,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_3794_power__inject__exp,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_3795_power__inject__exp,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_3796_power__inject__exp,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_3797_power__0__Suc,axiom,
    ! [Na: nat] : ( aa_nat_uint32(power_power_uint32(zero_zero_uint32),aa_nat_nat(suc,Na)) = zero_zero_uint32 ) ).

% power_0_Suc
tff(fact_3798_power__0__Suc,axiom,
    ! [Na: nat] : ( aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Na)) = zero_zero_rat ) ).

% power_0_Suc
tff(fact_3799_power__0__Suc,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Na)) = zero_zero_nat ) ).

% power_0_Suc
tff(fact_3800_power__0__Suc,axiom,
    ! [Na: nat] : ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Na)) = zero_zero_real ) ).

% power_0_Suc
tff(fact_3801_power__0__Suc,axiom,
    ! [Na: nat] : ( aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Na)) = zero_zero_complex ) ).

% power_0_Suc
tff(fact_3802_power__0__Suc,axiom,
    ! [Na: nat] : ( aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Na)) = zero_zero_int ) ).

% power_0_Suc
tff(fact_3803_power__Suc0__right,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,zero_zero_nat)) = A ) ).

% power_Suc0_right
tff(fact_3804_power__Suc0__right,axiom,
    ! [A: real] : ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ) ).

% power_Suc0_right
tff(fact_3805_power__Suc0__right,axiom,
    ! [A: complex] : ( aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,zero_zero_nat)) = A ) ).

% power_Suc0_right
tff(fact_3806_power__Suc0__right,axiom,
    ! [A: int] : ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,zero_zero_nat)) = A ) ).

% power_Suc0_right
tff(fact_3807_card_Oempty,axiom,
    finite_card_complex(bot_bot_set_complex) = zero_zero_nat ).

% card.empty
tff(fact_3808_card_Oempty,axiom,
    finite_card_literal(bot_bot_set_literal) = zero_zero_nat ).

% card.empty
tff(fact_3809_card_Oempty,axiom,
    finite_card_list_nat(bot_bot_set_list_nat) = zero_zero_nat ).

% card.empty
tff(fact_3810_card_Oempty,axiom,
    finite_card_set_nat(bot_bot_set_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_3811_card_Oempty,axiom,
    aa_set_nat_nat(finite_card_nat,bot_bot_set_nat) = zero_zero_nat ).

% card.empty
tff(fact_3812_card_Oempty,axiom,
    finite_card_int(bot_bot_set_int) = zero_zero_nat ).

% card.empty
tff(fact_3813_card_Oempty,axiom,
    finite_card_o(bot_bot_set_o) = zero_zero_nat ).

% card.empty
tff(fact_3814_card_Oinfinite,axiom,
    ! [A4: set_literal] :
      ( ~ finite5847741373460823677iteral(A4)
     => ( finite_card_literal(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_3815_card_Oinfinite,axiom,
    ! [A4: set_list_nat] :
      ( ~ finite8100373058378681591st_nat(A4)
     => ( finite_card_list_nat(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_3816_card_Oinfinite,axiom,
    ! [A4: set_set_nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( finite_card_set_nat(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_3817_card_Oinfinite,axiom,
    ! [A4: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(finite_card_nat,A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_3818_card_Oinfinite,axiom,
    ! [A4: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( finite_card_int(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_3819_card_Oinfinite,axiom,
    ! [A4: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( finite_card_complex(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_3820_card_Oinfinite,axiom,
    ! [A4: set_Code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( finite4902975817058060853nteger(A4) = zero_zero_nat ) ) ).

% card.infinite
tff(fact_3821_Gcd__fin_Oempty,axiom,
    aa_set_nat_nat(semiri4258706085729940814in_nat,bot_bot_set_nat) = zero_zero_nat ).

% Gcd_fin.empty
tff(fact_3822_Gcd__fin_Oempty,axiom,
    semiri4256215615220890538in_int(bot_bot_set_int) = zero_zero_int ).

% Gcd_fin.empty
tff(fact_3823_Gcd__fin_Oinfinite,axiom,
    ! [A4: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(semiri4258706085729940814in_nat,A4) = one_one_nat ) ) ).

% Gcd_fin.infinite
tff(fact_3824_Gcd__fin_Oinfinite,axiom,
    ! [A4: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( semiri4256215615220890538in_int(A4) = one_one_int ) ) ).

% Gcd_fin.infinite
tff(fact_3825_Gcd__fin__eq__Gcd,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(semiri4258706085729940814in_nat,A4) = aa_set_nat_nat(gcd_Gcd_nat,A4) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_3826_Gcd__fin__eq__Gcd,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( semiri4256215615220890538in_int(A4) = gcd_Gcd_int(A4) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_3827_power__strict__increasing__iff,axiom,
    ! [B: real,Xa: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3828_power__strict__increasing__iff,axiom,
    ! [B: rat,Xa: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Xa)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3829_power__strict__increasing__iff,axiom,
    ! [B: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3830_power__strict__increasing__iff,axiom,
    ! [B: int,Xa: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),Xa)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3831_power__eq__0__iff,axiom,
    ! [A: rat,Na: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3832_power__eq__0__iff,axiom,
    ! [A: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3833_power__eq__0__iff,axiom,
    ! [A: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3834_power__eq__0__iff,axiom,
    ! [A: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Na) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3835_power__eq__0__iff,axiom,
    ! [A: int,Na: nat] :
      ( ( aa_nat_int(power_power_int(A),Na) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3836_card__0__eq,axiom,
    ! [A4: set_literal] :
      ( finite5847741373460823677iteral(A4)
     => ( ( finite_card_literal(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_literal ) ) ) ).

% card_0_eq
tff(fact_3837_card__0__eq,axiom,
    ! [A4: set_list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( ( finite_card_list_nat(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
tff(fact_3838_card__0__eq,axiom,
    ! [A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( ( finite_card_set_nat(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
tff(fact_3839_card__0__eq,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( finite_card_complex(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_complex ) ) ) ).

% card_0_eq
tff(fact_3840_card__0__eq,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( finite4902975817058060853nteger(A4) = zero_zero_nat )
      <=> ( A4 = bot_bo3990330152332043303nteger ) ) ) ).

% card_0_eq
tff(fact_3841_card__0__eq,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(finite_card_nat,A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_nat ) ) ) ).

% card_0_eq
tff(fact_3842_card__0__eq,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( finite_card_int(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_int ) ) ) ).

% card_0_eq
tff(fact_3843_card__0__eq,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( finite_card_o(A4) = zero_zero_nat )
      <=> ( A4 = bot_bot_set_o ) ) ) ).

% card_0_eq
tff(fact_3844_card__insert__disjoint,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((Xa)),A4)
       => ( finite_card_o(aa_set_o_set_o(insert_o((Xa)),A4)) = aa_nat_nat(suc,finite_card_o(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3845_card__insert__disjoint,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( finite7802652506058667612T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = aa_nat_nat(suc,finite7802652506058667612T_VEBT(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3846_card__insert__disjoint,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(Xa),A4)
       => ( finite_card_real(aa_set_real_set_real(insert_real(Xa),A4)) = aa_nat_nat(suc,finite_card_real(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3847_card__insert__disjoint,axiom,
    ! [A4: set_literal,Xa: literal] :
      ( finite5847741373460823677iteral(A4)
     => ( ~ member_literal(Xa,A4)
       => ( finite_card_literal(insert_literal(Xa,A4)) = aa_nat_nat(suc,finite_card_literal(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3848_card__insert__disjoint,axiom,
    ! [A4: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( ~ member_list_nat(Xa,A4)
       => ( finite_card_list_nat(insert_list_nat(Xa,A4)) = aa_nat_nat(suc,finite_card_list_nat(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3849_card__insert__disjoint,axiom,
    ! [A4: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( ~ aa_set_set_nat_o(member_set_nat(Xa),A4)
       => ( finite_card_set_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4)) = aa_nat_nat(suc,finite_card_set_nat(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3850_card__insert__disjoint,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(insert_nat(Xa),A4)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3851_card__insert__disjoint,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(Xa),A4)
       => ( finite_card_int(aa_set_int_set_int(insert_int(Xa),A4)) = aa_nat_nat(suc,finite_card_int(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3852_card__insert__disjoint,axiom,
    ! [A4: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ~ aa_set_complex_o(member_complex(Xa),A4)
       => ( finite_card_complex(insert_complex(Xa,A4)) = aa_nat_nat(suc,finite_card_complex(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3853_card__insert__disjoint,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( finite4902975817058060853nteger(insert_Code_integer(Xa,A4)) = aa_nat_nat(suc,finite4902975817058060853nteger(A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_3854_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3855_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3856_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3857_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3858_power__mono__iff,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_3859_power__mono__iff,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_3860_power__mono__iff,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_3861_power__mono__iff,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_3862_power__increasing__iff,axiom,
    ! [B: real,Xa: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_3863_power__increasing__iff,axiom,
    ! [B: rat,Xa: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Xa)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_3864_power__increasing__iff,axiom,
    ! [B: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_3865_power__increasing__iff,axiom,
    ! [B: int,Xa: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),Xa)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ) ).

% power_increasing_iff
tff(fact_3866_card__Diff__insert,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( ~ aa_set_o_o(member_o((A)),B4)
       => ( finite_card_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(minus_minus_set_o(A4,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_3867_card__Diff__insert,axiom,
    ! [A: vEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),B4)
       => ( finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_3868_card__Diff__insert,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( aa_set_real_o(member_real(A),A4)
     => ( ~ aa_set_real_o(member_real(A),B4)
       => ( finite_card_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(minus_minus_set_real(A4,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_3869_card__Diff__insert,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( ~ aa_set_int_o(member_int(A),B4)
       => ( finite_card_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(minus_minus_set_int(A4,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_3870_card__Diff__insert,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(member_complex(A),A4)
     => ( ~ aa_set_complex_o(member_complex(A),B4)
       => ( finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(A,B4))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(minus_811609699411566653omplex(A4,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_3871_card__Diff__insert,axiom,
    ! [A: literal,A4: set_literal,B4: set_literal] :
      ( member_literal(A,A4)
     => ( ~ member_literal(A,B4)
       => ( finite_card_literal(minus_7832829386415567259iteral(A4,insert_literal(A,B4))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_literal(minus_7832829386415567259iteral(A4,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_3872_card__Diff__insert,axiom,
    ! [A: list_nat,A4: set_list_nat,B4: set_list_nat] :
      ( member_list_nat(A,A4)
     => ( ~ member_list_nat(A,B4)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(A,B4))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_3873_card__Diff__insert,axiom,
    ! [A: set_nat,A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(A),A4)
     => ( ~ aa_set_set_nat_o(member_set_nat(A),B4)
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_3874_card__Diff__insert,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( ~ aa_set_nat_o(member_nat(A),B4)
       => ( aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(A),B4))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,B4))),one_one_nat) ) ) ) ).

% card_Diff_insert
tff(fact_3875_finite__enumerate__mono__iff,axiom,
    ! [S2: set_nat,M: nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_set_nat_nat(finite_card_nat,S2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_set_nat_nat(finite_card_nat,S2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S2),M)),aa_nat_nat(infini8530281810654367211te_nat(S2),Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_3876_size__neq__size__imp__neq,axiom,
    ! [Xa: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xa) != size_s6755466524823107622T_VEBT(Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_3877_size__neq__size__imp__neq,axiom,
    ! [Xa: list_real,Y: list_real] :
      ( ( size_size_list_real(Xa) != size_size_list_real(Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_3878_size__neq__size__imp__neq,axiom,
    ! [Xa: list_o,Y: list_o] :
      ( ( size_size_list_o(Xa) != size_size_list_o(Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_3879_size__neq__size__imp__neq,axiom,
    ! [Xa: list_nat,Y: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xa) != aa_list_nat_nat(size_size_list_nat,Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_3880_size__neq__size__imp__neq,axiom,
    ! [Xa: num,Y: num] :
      ( ( aa_num_nat(size_size_num,Xa) != aa_num_nat(size_size_num,Y) )
     => ( Xa != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_3881_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),Xa)),size_s6755466524823107622T_VEBT(Xs)) ).

% count_le_length
tff(fact_3882_count__le__length,axiom,
    ! [Xs: list_real,Xa: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(count_list_real(Xs),Xa)),size_size_list_real(Xs)) ).

% count_le_length
tff(fact_3883_count__le__length,axiom,
    ! [Xs: list_o,Xa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(count_list_o(Xs),(Xa))),size_size_list_o(Xs)) ).

% count_le_length
tff(fact_3884_count__le__length,axiom,
    ! [Xs: list_nat,Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(count_list_nat(Xs),Xa)),aa_list_nat_nat(size_size_list_nat,Xs)) ).

% count_le_length
tff(fact_3885_card__length,axiom,
    ! [Xs: list_complex] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(set_complex2(Xs))),size_s3451745648224563538omplex(Xs)) ).

% card_length
tff(fact_3886_card__length,axiom,
    ! [Xs: list_literal] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(set_literal2(Xs))),size_s2501651207091587910iteral(Xs)) ).

% card_length
tff(fact_3887_card__length,axiom,
    ! [Xs: list_list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(set_list_nat2(Xs))),size_s3023201423986296836st_nat(Xs)) ).

% card_length
tff(fact_3888_card__length,axiom,
    ! [Xs: list_set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(set_set_nat2(Xs))),size_s3254054031482475050et_nat(Xs)) ).

% card_length
tff(fact_3889_card__length,axiom,
    ! [Xs: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))),size_s6755466524823107622T_VEBT(Xs)) ).

% card_length
tff(fact_3890_card__length,axiom,
    ! [Xs: list_real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(aa_lis3769085826154824284t_real(set_real2,Xs))),size_size_list_real(Xs)) ).

% card_length
tff(fact_3891_card__length,axiom,
    ! [Xs: list_o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(aa_list_o_set_o(set_o2,Xs))),size_size_list_o(Xs)) ).

% card_length
tff(fact_3892_card__length,axiom,
    ! [Xs: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_list_nat_set_nat(set_nat2,Xs))),aa_list_nat_nat(size_size_list_nat,Xs)) ).

% card_length
tff(fact_3893_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,Na: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Na) != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_3894_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: nat,Na: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_3895_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: real,Na: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_3896_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: complex,Na: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Na) != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_3897_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,Na: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Na) != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_3898_length__induct,axiom,
    ! [P2: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P2,Ys2) )
         => aa_list_VEBT_VEBT_o(P2,Xs2) )
     => aa_list_VEBT_VEBT_o(P2,Xs) ) ).

% length_induct
tff(fact_3899_length__induct,axiom,
    ! [P2: fun_list_real_o,Xs: list_real] :
      ( ! [Xs2: list_real] :
          ( ! [Ys2: list_real] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(Ys2)),size_size_list_real(Xs2))
             => aa_list_real_o(P2,Ys2) )
         => aa_list_real_o(P2,Xs2) )
     => aa_list_real_o(P2,Xs) ) ).

% length_induct
tff(fact_3900_length__induct,axiom,
    ! [P2: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P2,Ys2) )
         => aa_list_o_o(P2,Xs2) )
     => aa_list_o_o(P2,Xs) ) ).

% length_induct
tff(fact_3901_length__induct,axiom,
    ! [P2: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ys2)),aa_list_nat_nat(size_size_list_nat,Xs2))
             => aa_list_nat_o(P2,Ys2) )
         => aa_list_nat_o(P2,Xs2) )
     => aa_list_nat_o(P2,Xs) ) ).

% length_induct
tff(fact_3902_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M7)
     => ? [N: nat] :
        ! [X3: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X3,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X3)),N) ) ) ).

% finite_maxlen
tff(fact_3903_finite__maxlen,axiom,
    ! [M7: set_list_real] :
      ( finite306553202115118035t_real(M7)
     => ? [N: nat] :
        ! [X3: list_real] :
          ( member_list_real(X3,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(X3)),N) ) ) ).

% finite_maxlen
tff(fact_3904_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( finite_finite_list_o(M7)
     => ? [N: nat] :
        ! [X3: list_o] :
          ( member_list_o(X3,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X3)),N) ) ) ).

% finite_maxlen
tff(fact_3905_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( finite8100373058378681591st_nat(M7)
     => ? [N: nat] :
        ! [X3: list_nat] :
          ( member_list_nat(X3,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,X3)),N) ) ) ).

% finite_maxlen
tff(fact_3906_zero__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_le_power
tff(fact_3907_zero__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_le_power
tff(fact_3908_zero__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_le_power
tff(fact_3909_zero__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_le_power
tff(fact_3910_power__mono,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ).

% power_mono
tff(fact_3911_power__mono,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ).

% power_mono
tff(fact_3912_power__mono,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ).

% power_mono
tff(fact_3913_power__mono,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ).

% power_mono
tff(fact_3914_zero__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% zero_less_power
tff(fact_3915_zero__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% zero_less_power
tff(fact_3916_zero__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% zero_less_power
tff(fact_3917_zero__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Na)) ) ).

% zero_less_power
tff(fact_3918_one__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% one_le_power
tff(fact_3919_one__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ).

% one_le_power
tff(fact_3920_one__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% one_le_power
tff(fact_3921_one__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ).

% one_le_power
tff(fact_3922_card__subset__eq,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( finite5847741373460823677iteral(B4)
     => ( ord_le7307670543136651348iteral(A4,B4)
       => ( ( finite_card_literal(A4) = finite_card_literal(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_3923_card__subset__eq,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le6045566169113846134st_nat(A4,B4)
       => ( ( finite_card_list_nat(A4) = finite_card_list_nat(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_3924_card__subset__eq,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
       => ( ( finite_card_set_nat(A4) = finite_card_set_nat(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_3925_card__subset__eq,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_set_nat_nat(finite_card_nat,B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_3926_card__subset__eq,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ( finite_card_complex(A4) = finite_card_complex(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_3927_card__subset__eq,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ( finite4902975817058060853nteger(A4) = finite4902975817058060853nteger(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_3928_card__subset__eq,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( ( finite_card_int(A4) = finite_card_int(B4) )
         => ( A4 = B4 ) ) ) ) ).

% card_subset_eq
tff(fact_3929_infinite__arbitrarily__large,axiom,
    ! [A4: set_literal,Na: nat] :
      ( ~ finite5847741373460823677iteral(A4)
     => ? [B7: set_literal] :
          ( finite5847741373460823677iteral(B7)
          & ( finite_card_literal(B7) = Na )
          & ord_le7307670543136651348iteral(B7,A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_3930_infinite__arbitrarily__large,axiom,
    ! [A4: set_list_nat,Na: nat] :
      ( ~ finite8100373058378681591st_nat(A4)
     => ? [B7: set_list_nat] :
          ( finite8100373058378681591st_nat(B7)
          & ( finite_card_list_nat(B7) = Na )
          & ord_le6045566169113846134st_nat(B7,A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_3931_infinite__arbitrarily__large,axiom,
    ! [A4: set_set_nat,Na: nat] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ? [B7: set_set_nat] :
          ( aa_set_set_nat_o(finite1152437895449049373et_nat,B7)
          & ( finite_card_set_nat(B7) = Na )
          & aa_set_set_nat_o(ord_le6893508408891458716et_nat(B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_3932_infinite__arbitrarily__large,axiom,
    ! [A4: set_nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ? [B7: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B7)
          & ( aa_set_nat_nat(finite_card_nat,B7) = Na )
          & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_3933_infinite__arbitrarily__large,axiom,
    ! [A4: set_complex,Na: nat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ? [B7: set_complex] :
          ( aa_set_complex_o(finite3207457112153483333omplex,B7)
          & ( finite_card_complex(B7) = Na )
          & aa_set_complex_o(ord_le211207098394363844omplex(B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_3934_infinite__arbitrarily__large,axiom,
    ! [A4: set_Code_integer,Na: nat] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ? [B7: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B7)
          & ( finite4902975817058060853nteger(B7) = Na )
          & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_3935_infinite__arbitrarily__large,axiom,
    ! [A4: set_int,Na: nat] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ? [B7: set_int] :
          ( aa_set_int_o(finite_finite_int,B7)
          & ( finite_card_int(B7) = Na )
          & aa_set_int_o(ord_less_eq_set_int(B7),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_3936_card__le__if__inj__on__rel,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,R: fun_VE2962686792283380846VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( ! [A3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A4)
           => ? [B9: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B9),B4)
                & aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R,A3),B9) ) )
       => ( ! [A1: vEBT_VEBT,A22: vEBT_VEBT,B3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A1),A4)
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A22),A4)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),B4)
                 => ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R,A1),B3)
                   => ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(A4)),finite7802652506058667612T_VEBT(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3937_card__le__if__inj__on__rel,axiom,
    ! [B4: set_real,A4: set_VEBT_VEBT,R: fun_VE3823611782367660756real_o] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( ! [A3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A4)
           => ? [B9: real] :
                ( aa_set_real_o(member_real(B9),B4)
                & aa_real_o(aa_VEB641519613517496269real_o(R,A3),B9) ) )
       => ( ! [A1: vEBT_VEBT,A22: vEBT_VEBT,B3: real] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A1),A4)
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A22),A4)
               => ( aa_set_real_o(member_real(B3),B4)
                 => ( aa_real_o(aa_VEB641519613517496269real_o(R,A1),B3)
                   => ( aa_real_o(aa_VEB641519613517496269real_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(A4)),finite_card_real(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3938_card__le__if__inj__on__rel,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_real,R: fun_re7829509182167823952VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A4)
           => ? [B9: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B9),B4)
                & aa_VEBT_VEBT_o(aa_rea6225614514526338833VEBT_o(R,A3),B9) ) )
       => ( ! [A1: real,A22: real,B3: vEBT_VEBT] :
              ( aa_set_real_o(member_real(A1),A4)
             => ( aa_set_real_o(member_real(A22),A4)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),B4)
                 => ( aa_VEBT_VEBT_o(aa_rea6225614514526338833VEBT_o(R,A1),B3)
                   => ( aa_VEBT_VEBT_o(aa_rea6225614514526338833VEBT_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A4)),finite7802652506058667612T_VEBT(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3939_card__le__if__inj__on__rel,axiom,
    ! [B4: set_real,A4: set_real,R: fun_real_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A4)
           => ? [B9: real] :
                ( aa_set_real_o(member_real(B9),B4)
                & aa_real_o(aa_real_fun_real_o(R,A3),B9) ) )
       => ( ! [A1: real,A22: real,B3: real] :
              ( aa_set_real_o(member_real(A1),A4)
             => ( aa_set_real_o(member_real(A22),A4)
               => ( aa_set_real_o(member_real(B3),B4)
                 => ( aa_real_o(aa_real_fun_real_o(R,A1),B3)
                   => ( aa_real_o(aa_real_fun_real_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A4)),finite_card_real(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3940_card__le__if__inj__on__rel,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_int,R: fun_in4578122135805219920VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ? [B9: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B9),B4)
                & aa_VEBT_VEBT_o(aa_int8839263053405020433VEBT_o(R,A3),B9) ) )
       => ( ! [A1: int,A22: int,B3: vEBT_VEBT] :
              ( aa_set_int_o(member_int(A1),A4)
             => ( aa_set_int_o(member_int(A22),A4)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),B4)
                 => ( aa_VEBT_VEBT_o(aa_int8839263053405020433VEBT_o(R,A1),B3)
                   => ( aa_VEBT_VEBT_o(aa_int8839263053405020433VEBT_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite7802652506058667612T_VEBT(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3941_card__le__if__inj__on__rel,axiom,
    ! [B4: set_real,A4: set_int,R: fun_int_fun_real_o] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ? [B9: real] :
                ( aa_set_real_o(member_real(B9),B4)
                & aa_real_o(aa_int_fun_real_o(R,A3),B9) ) )
       => ( ! [A1: int,A22: int,B3: real] :
              ( aa_set_int_o(member_int(A1),A4)
             => ( aa_set_int_o(member_int(A22),A4)
               => ( aa_set_real_o(member_real(B3),B4)
                 => ( aa_real_o(aa_int_fun_real_o(R,A1),B3)
                   => ( aa_real_o(aa_int_fun_real_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_real(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3942_card__le__if__inj__on__rel,axiom,
    ! [B4: set_literal,A4: set_VEBT_VEBT,R: fun_VE8371440389858774350eral_o] :
      ( finite5847741373460823677iteral(B4)
     => ( ! [A3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A4)
           => ? [B9: literal] :
                ( member_literal(B9,B4)
                & aa_literal_o(aa_VEB1719478621182999551eral_o(R,A3),B9) ) )
       => ( ! [A1: vEBT_VEBT,A22: vEBT_VEBT,B3: literal] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A1),A4)
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A22),A4)
               => ( member_literal(B3,B4)
                 => ( aa_literal_o(aa_VEB1719478621182999551eral_o(R,A1),B3)
                   => ( aa_literal_o(aa_VEB1719478621182999551eral_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(A4)),finite_card_literal(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3943_card__le__if__inj__on__rel,axiom,
    ! [B4: set_literal,A4: set_real,R: fun_re3850508403959675952eral_o] :
      ( finite5847741373460823677iteral(B4)
     => ( ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A4)
           => ? [B9: literal] :
                ( member_literal(B9,B4)
                & aa_literal_o(aa_rea1838623199774720753eral_o(R,A3),B9) ) )
       => ( ! [A1: real,A22: real,B3: literal] :
              ( aa_set_real_o(member_real(A1),A4)
             => ( aa_set_real_o(member_real(A22),A4)
               => ( member_literal(B3,B4)
                 => ( aa_literal_o(aa_rea1838623199774720753eral_o(R,A1),B3)
                   => ( aa_literal_o(aa_rea1838623199774720753eral_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_real(A4)),finite_card_literal(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3944_card__le__if__inj__on__rel,axiom,
    ! [B4: set_literal,A4: set_int,R: fun_in476721220626635312eral_o] :
      ( finite5847741373460823677iteral(B4)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ? [B9: literal] :
                ( member_literal(B9,B4)
                & aa_literal_o(aa_int_fun_literal_o(R,A3),B9) ) )
       => ( ! [A1: int,A22: int,B3: literal] :
              ( aa_set_int_o(member_int(A1),A4)
             => ( aa_set_int_o(member_int(A22),A4)
               => ( member_literal(B3,B4)
                 => ( aa_literal_o(aa_int_fun_literal_o(R,A1),B3)
                   => ( aa_literal_o(aa_int_fun_literal_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_literal(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3945_card__le__if__inj__on__rel,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_complex,R: fun_co9003232526696249554VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( ! [A3: complex] :
            ( aa_set_complex_o(member_complex(A3),A4)
           => ? [B9: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B9),B4)
                & aa_VEBT_VEBT_o(aa_com3651963455866140051VEBT_o(R,A3),B9) ) )
       => ( ! [A1: complex,A22: complex,B3: vEBT_VEBT] :
              ( aa_set_complex_o(member_complex(A1),A4)
             => ( aa_set_complex_o(member_complex(A22),A4)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),B4)
                 => ( aa_VEBT_VEBT_o(aa_com3651963455866140051VEBT_o(R,A1),B3)
                   => ( aa_VEBT_VEBT_o(aa_com3651963455866140051VEBT_o(R,A22),B3)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A4)),finite7802652506058667612T_VEBT(B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3946_empty__in__Fpow,axiom,
    ! [A4: set_nat] : aa_set_set_nat_o(member_set_nat(bot_bot_set_nat),finite_Fpow_nat(A4)) ).

% empty_in_Fpow
tff(fact_3947_empty__in__Fpow,axiom,
    ! [A4: set_int] : member_set_int(bot_bot_set_int,finite_Fpow_int(A4)) ).

% empty_in_Fpow
tff(fact_3948_empty__in__Fpow,axiom,
    ! [A4: set_o] : member_set_o(bot_bot_set_o,finite_Fpow_o(A4)) ).

% empty_in_Fpow
tff(fact_3949_card__insert__le,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(A4)),finite7802652506058667612T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4))) ).

% card_insert_le
tff(fact_3950_card__insert__le,axiom,
    ! [A4: set_int,Xa: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(aa_set_int_set_int(insert_int(Xa),A4))) ).

% card_insert_le
tff(fact_3951_card__insert__le,axiom,
    ! [A4: set_o,Xa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(A4)),finite_card_o(aa_set_o_set_o(insert_o((Xa)),A4))) ).

% card_insert_le
tff(fact_3952_card__insert__le,axiom,
    ! [A4: set_complex,Xa: complex] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A4)),finite_card_complex(insert_complex(Xa,A4))) ).

% card_insert_le
tff(fact_3953_card__insert__le,axiom,
    ! [A4: set_literal,Xa: literal] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(A4)),finite_card_literal(insert_literal(Xa,A4))) ).

% card_insert_le
tff(fact_3954_card__insert__le,axiom,
    ! [A4: set_list_nat,Xa: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A4)),finite_card_list_nat(insert_list_nat(Xa,A4))) ).

% card_insert_le
tff(fact_3955_card__insert__le,axiom,
    ! [A4: set_set_nat,Xa: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A4)),finite_card_set_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4))) ).

% card_insert_le
tff(fact_3956_card__insert__le,axiom,
    ! [A4: set_nat,Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(insert_nat(Xa),A4))) ).

% card_insert_le
tff(fact_3957_power__Suc,axiom,
    ! [A: complex,Na: nat] : ( aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Na)) ) ).

% power_Suc
tff(fact_3958_power__Suc,axiom,
    ! [A: assn,Na: nat] : ( aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,Na)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),Na)) ) ).

% power_Suc
tff(fact_3959_power__Suc,axiom,
    ! [A: uint32,Na: nat] : ( aa_nat_uint32(power_power_uint32(A),aa_nat_nat(suc,Na)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,A),aa_nat_uint32(power_power_uint32(A),Na)) ) ).

% power_Suc
tff(fact_3960_power__Suc,axiom,
    ! [A: real,Na: nat] : ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ).

% power_Suc
tff(fact_3961_power__Suc,axiom,
    ! [A: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ).

% power_Suc
tff(fact_3962_power__Suc,axiom,
    ! [A: int,Na: nat] : ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na)) ) ).

% power_Suc
tff(fact_3963_power__Suc2,axiom,
    ! [A: complex,Na: nat] : ( aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Na)),A) ) ).

% power_Suc2
tff(fact_3964_power__Suc2,axiom,
    ! [A: assn,Na: nat] : ( aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,Na)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),Na)),A) ) ).

% power_Suc2
tff(fact_3965_power__Suc2,axiom,
    ! [A: uint32,Na: nat] : ( aa_nat_uint32(power_power_uint32(A),aa_nat_nat(suc,Na)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(power_power_uint32(A),Na)),A) ) ).

% power_Suc2
tff(fact_3966_power__Suc2,axiom,
    ! [A: real,Na: nat] : ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),A) ) ).

% power_Suc2
tff(fact_3967_power__Suc2,axiom,
    ! [A: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),A) ) ).

% power_Suc2
tff(fact_3968_power__Suc2,axiom,
    ! [A: int,Na: nat] : ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Na)),A) ) ).

% power_Suc2
tff(fact_3969_Fpow__mono,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
     => ord_le4403425263959731960et_int(finite_Fpow_int(A4),finite_Fpow_int(B4)) ) ).

% Fpow_mono
tff(fact_3970_power__0,axiom,
    ! [A: uint32] : ( aa_nat_uint32(power_power_uint32(A),zero_zero_nat) = one_one_uint32 ) ).

% power_0
tff(fact_3971_power__0,axiom,
    ! [A: rat] : ( aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ) ).

% power_0
tff(fact_3972_power__0,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ) ).

% power_0
tff(fact_3973_power__0,axiom,
    ! [A: real] : ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ) ).

% power_0
tff(fact_3974_power__0,axiom,
    ! [A: complex] : ( aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ) ).

% power_0
tff(fact_3975_power__0,axiom,
    ! [A: int] : ( aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ) ).

% power_0
tff(fact_3976_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_3977_length__removeAll__less__eq,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(removeAll_VEBT_VEBT(Xa,Xs))),size_s6755466524823107622T_VEBT(Xs)) ).

% length_removeAll_less_eq
tff(fact_3978_length__removeAll__less__eq,axiom,
    ! [Xa: real,Xs: list_real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(removeAll_real(Xa,Xs))),size_size_list_real(Xs)) ).

% length_removeAll_less_eq
tff(fact_3979_length__removeAll__less__eq,axiom,
    ! [Xa: $o,Xs: list_o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(removeAll_o((Xa),Xs))),size_size_list_o(Xs)) ).

% length_removeAll_less_eq
tff(fact_3980_length__removeAll__less__eq,axiom,
    ! [Xa: nat,Xs: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,removeAll_nat(Xa,Xs))),aa_list_nat_nat(size_size_list_nat,Xs)) ).

% length_removeAll_less_eq
tff(fact_3981_is__singleton__altdef,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(is_singleton_complex,A4)
    <=> ( finite_card_complex(A4) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3982_is__singleton__altdef,axiom,
    ! [A4: set_literal] :
      ( is_singleton_literal(A4)
    <=> ( finite_card_literal(A4) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3983_is__singleton__altdef,axiom,
    ! [A4: set_list_nat] :
      ( is_sin2641923865335537900st_nat(A4)
    <=> ( finite_card_list_nat(A4) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3984_is__singleton__altdef,axiom,
    ! [A4: set_set_nat] :
      ( aa_set_set_nat_o(is_singleton_set_nat,A4)
    <=> ( finite_card_set_nat(A4) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3985_is__singleton__altdef,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(is_singleton_nat,A4)
    <=> ( aa_set_nat_nat(finite_card_nat,A4) = one_one_nat ) ) ).

% is_singleton_altdef
tff(fact_3986_power__less__imp__less__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3987_power__less__imp__less__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3988_power__less__imp__less__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3989_power__less__imp__less__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3990_card__ge__0__finite,axiom,
    ! [A4: set_literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_literal(A4))
     => finite5847741373460823677iteral(A4) ) ).

% card_ge_0_finite
tff(fact_3991_card__ge__0__finite,axiom,
    ! [A4: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_list_nat(A4))
     => finite8100373058378681591st_nat(A4) ) ).

% card_ge_0_finite
tff(fact_3992_card__ge__0__finite,axiom,
    ! [A4: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_set_nat(A4))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,A4) ) ).

% card_ge_0_finite
tff(fact_3993_card__ge__0__finite,axiom,
    ! [A4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A4))
     => aa_set_nat_o(finite_finite_nat,A4) ) ).

% card_ge_0_finite
tff(fact_3994_card__ge__0__finite,axiom,
    ! [A4: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A4))
     => aa_set_int_o(finite_finite_int,A4) ) ).

% card_ge_0_finite
tff(fact_3995_card__ge__0__finite,axiom,
    ! [A4: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_complex(A4))
     => aa_set_complex_o(finite3207457112153483333omplex,A4) ) ).

% card_ge_0_finite
tff(fact_3996_card__ge__0__finite,axiom,
    ! [A4: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite4902975817058060853nteger(A4))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ).

% card_ge_0_finite
tff(fact_3997_card__eq__0__iff,axiom,
    ! [A4: set_literal] :
      ( ( finite_card_literal(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_literal )
        | ~ finite5847741373460823677iteral(A4) ) ) ).

% card_eq_0_iff
tff(fact_3998_card__eq__0__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( finite_card_list_nat(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_list_nat )
        | ~ finite8100373058378681591st_nat(A4) ) ) ).

% card_eq_0_iff
tff(fact_3999_card__eq__0__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( finite_card_set_nat(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_set_nat )
        | ~ aa_set_set_nat_o(finite1152437895449049373et_nat,A4) ) ) ).

% card_eq_0_iff
tff(fact_4000_card__eq__0__iff,axiom,
    ! [A4: set_complex] :
      ( ( finite_card_complex(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_complex )
        | ~ aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% card_eq_0_iff
tff(fact_4001_card__eq__0__iff,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite4902975817058060853nteger(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bo3990330152332043303nteger )
        | ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ) ).

% card_eq_0_iff
tff(fact_4002_card__eq__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_nat )
        | ~ aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% card_eq_0_iff
tff(fact_4003_card__eq__0__iff,axiom,
    ! [A4: set_int] :
      ( ( finite_card_int(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_int )
        | ~ aa_set_int_o(finite_finite_int,A4) ) ) ).

% card_eq_0_iff
tff(fact_4004_card__eq__0__iff,axiom,
    ! [A4: set_o] :
      ( ( finite_card_o(A4) = zero_zero_nat )
    <=> ( ( A4 = bot_bot_set_o )
        | ~ aa_set_o_o(finite_finite_o,A4) ) ) ).

% card_eq_0_iff
tff(fact_4005_power__le__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),one_one_real) ) ) ).

% power_le_one
tff(fact_4006_power__le__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),one_one_rat) ) ) ).

% power_le_one
tff(fact_4007_power__le__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),one_one_nat) ) ) ).

% power_le_one
tff(fact_4008_power__le__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),one_one_int) ) ) ).

% power_le_one
tff(fact_4009_card__insert__if,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( finite_card_o(aa_set_o_set_o(insert_o((Xa)),A4)) = $ite(aa_set_o_o(member_o((Xa)),A4),finite_card_o(A4),aa_nat_nat(suc,finite_card_o(A4))) ) ) ).

% card_insert_if
tff(fact_4010_card__insert__if,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( finite7802652506058667612T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4),finite7802652506058667612T_VEBT(A4),aa_nat_nat(suc,finite7802652506058667612T_VEBT(A4))) ) ) ).

% card_insert_if
tff(fact_4011_card__insert__if,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( finite_card_real(aa_set_real_set_real(insert_real(Xa),A4)) = $ite(aa_set_real_o(member_real(Xa),A4),finite_card_real(A4),aa_nat_nat(suc,finite_card_real(A4))) ) ) ).

% card_insert_if
tff(fact_4012_card__insert__if,axiom,
    ! [A4: set_literal,Xa: literal] :
      ( finite5847741373460823677iteral(A4)
     => ( finite_card_literal(insert_literal(Xa,A4)) = $ite(member_literal(Xa,A4),finite_card_literal(A4),aa_nat_nat(suc,finite_card_literal(A4))) ) ) ).

% card_insert_if
tff(fact_4013_card__insert__if,axiom,
    ! [A4: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite_card_list_nat(insert_list_nat(Xa,A4)) = $ite(member_list_nat(Xa,A4),finite_card_list_nat(A4),aa_nat_nat(suc,finite_card_list_nat(A4))) ) ) ).

% card_insert_if
tff(fact_4014_card__insert__if,axiom,
    ! [A4: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( finite_card_set_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4)) = $ite(aa_set_set_nat_o(member_set_nat(Xa),A4),finite_card_set_nat(A4),aa_nat_nat(suc,finite_card_set_nat(A4))) ) ) ).

% card_insert_if
tff(fact_4015_card__insert__if,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(insert_nat(Xa),A4)) = $ite(aa_set_nat_o(member_nat(Xa),A4),aa_set_nat_nat(finite_card_nat,A4),aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,A4))) ) ) ).

% card_insert_if
tff(fact_4016_card__insert__if,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( finite_card_int(aa_set_int_set_int(insert_int(Xa),A4)) = $ite(aa_set_int_o(member_int(Xa),A4),finite_card_int(A4),aa_nat_nat(suc,finite_card_int(A4))) ) ) ).

% card_insert_if
tff(fact_4017_card__insert__if,axiom,
    ! [A4: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( finite_card_complex(insert_complex(Xa,A4)) = $ite(aa_set_complex_o(member_complex(Xa),A4),finite_card_complex(A4),aa_nat_nat(suc,finite_card_complex(A4))) ) ) ).

% card_insert_if
tff(fact_4018_card__insert__if,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( finite4902975817058060853nteger(insert_Code_integer(Xa,A4)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4),finite4902975817058060853nteger(A4),aa_nat_nat(suc,finite4902975817058060853nteger(A4))) ) ) ).

% card_insert_if
tff(fact_4019_card__Suc__eq__finite,axiom,
    ! [A4: set_o,K: nat] :
      ( ( finite_card_o(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: $o,B8: set_o] :
          ( ( A4 = aa_set_o_set_o(insert_o((B6)),B8) )
          & ~ aa_set_o_o(member_o((B6)),B8)
          & ( finite_card_o(B8) = K )
          & aa_set_o_o(finite_finite_o,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4020_card__Suc__eq__finite,axiom,
    ! [A4: set_VEBT_VEBT,K: nat] :
      ( ( finite7802652506058667612T_VEBT(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: vEBT_VEBT,B8: set_VEBT_VEBT] :
          ( ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B6),B8) )
          & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B6),B8)
          & ( finite7802652506058667612T_VEBT(B8) = K )
          & aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4021_card__Suc__eq__finite,axiom,
    ! [A4: set_real,K: nat] :
      ( ( finite_card_real(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: real,B8: set_real] :
          ( ( A4 = aa_set_real_set_real(insert_real(B6),B8) )
          & ~ aa_set_real_o(member_real(B6),B8)
          & ( finite_card_real(B8) = K )
          & aa_set_real_o(finite_finite_real,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4022_card__Suc__eq__finite,axiom,
    ! [A4: set_literal,K: nat] :
      ( ( finite_card_literal(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: literal,B8: set_literal] :
          ( ( A4 = insert_literal(B6,B8) )
          & ~ member_literal(B6,B8)
          & ( finite_card_literal(B8) = K )
          & finite5847741373460823677iteral(B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4023_card__Suc__eq__finite,axiom,
    ! [A4: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: list_nat,B8: set_list_nat] :
          ( ( A4 = insert_list_nat(B6,B8) )
          & ~ member_list_nat(B6,B8)
          & ( finite_card_list_nat(B8) = K )
          & finite8100373058378681591st_nat(B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4024_card__Suc__eq__finite,axiom,
    ! [A4: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: set_nat,B8: set_set_nat] :
          ( ( A4 = aa_set1013257548857649334et_nat(insert_set_nat(B6),B8) )
          & ~ aa_set_set_nat_o(member_set_nat(B6),B8)
          & ( finite_card_set_nat(B8) = K )
          & aa_set_set_nat_o(finite1152437895449049373et_nat,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4025_card__Suc__eq__finite,axiom,
    ! [A4: set_nat,K: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: nat,B8: set_nat] :
          ( ( A4 = aa_set_nat_set_nat(insert_nat(B6),B8) )
          & ~ aa_set_nat_o(member_nat(B6),B8)
          & ( aa_set_nat_nat(finite_card_nat,B8) = K )
          & aa_set_nat_o(finite_finite_nat,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4026_card__Suc__eq__finite,axiom,
    ! [A4: set_int,K: nat] :
      ( ( finite_card_int(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: int,B8: set_int] :
          ( ( A4 = aa_set_int_set_int(insert_int(B6),B8) )
          & ~ aa_set_int_o(member_int(B6),B8)
          & ( finite_card_int(B8) = K )
          & aa_set_int_o(finite_finite_int,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4027_card__Suc__eq__finite,axiom,
    ! [A4: set_complex,K: nat] :
      ( ( finite_card_complex(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: complex,B8: set_complex] :
          ( ( A4 = insert_complex(B6,B8) )
          & ~ aa_set_complex_o(member_complex(B6),B8)
          & ( finite_card_complex(B8) = K )
          & aa_set_complex_o(finite3207457112153483333omplex,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4028_card__Suc__eq__finite,axiom,
    ! [A4: set_Code_integer,K: nat] :
      ( ( finite4902975817058060853nteger(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: code_integer,B8: set_Code_integer] :
          ( ( A4 = insert_Code_integer(B6,B8) )
          & ~ aa_set5643387332196575111eger_o(member_Code_integer(B6),B8)
          & ( finite4902975817058060853nteger(B8) = K )
          & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8) ) ) ).

% card_Suc_eq_finite
tff(fact_4029_power__inject__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Na)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4030_power__inject__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Na)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4031_power__inject__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Na)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4032_power__inject__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Na)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_4033_power__le__imp__le__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Na)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_4034_power__le__imp__le__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Na)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_4035_power__le__imp__le__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Na)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_4036_power__le__imp__le__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Na)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_4037_card__image__le,axiom,
    ! [A4: set_literal,F: fun_literal_complex] :
      ( finite5847741373460823677iteral(A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(image_5274195009022015549omplex(F,A4))),finite_card_literal(A4)) ) ).

% card_image_le
tff(fact_4038_card__image__le,axiom,
    ! [A4: set_literal,F: fun_literal_literal] :
      ( finite5847741373460823677iteral(A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(image_8195128725298311301iteral(F,A4))),finite_card_literal(A4)) ) ).

% card_image_le
tff(fact_4039_card__image__le,axiom,
    ! [A4: set_literal,F: fun_literal_nat] :
      ( finite5847741373460823677iteral(A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,image_literal_nat(F,A4))),finite_card_literal(A4)) ) ).

% card_image_le
tff(fact_4040_card__image__le,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_nat_set_int(image_nat_int(F),A4))),aa_set_nat_nat(finite_card_nat,A4)) ) ).

% card_image_le
tff(fact_4041_card__image__le,axiom,
    ! [A4: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(image_nat_complex(F,A4))),aa_set_nat_nat(finite_card_nat,A4)) ) ).

% card_image_le
tff(fact_4042_card__image__le,axiom,
    ! [A4: set_nat,F: fun_nat_literal] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(image_nat_literal(F,A4))),aa_set_nat_nat(finite_card_nat,A4)) ) ).

% card_image_le
tff(fact_4043_card__image__le,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(finite_card_nat,A4)) ) ).

% card_image_le
tff(fact_4044_card__image__le,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(image_int_int(F),A4))),finite_card_int(A4)) ) ).

% card_image_le
tff(fact_4045_card__image__le,axiom,
    ! [A4: set_int,F: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(image_int_complex(F,A4))),finite_card_int(A4)) ) ).

% card_image_le
tff(fact_4046_card__image__le,axiom,
    ! [A4: set_int,F: fun_int_literal] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(image_int_literal(F,A4))),finite_card_int(A4)) ) ).

% card_image_le
tff(fact_4047_power__gt1__lemma,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))) ) ).

% power_gt1_lemma
tff(fact_4048_power__gt1__lemma,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_gt1_lemma
tff(fact_4049_power__gt1__lemma,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))) ) ).

% power_gt1_lemma
tff(fact_4050_power__gt1__lemma,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))) ) ).

% power_gt1_lemma
tff(fact_4051_power__less__power__Suc,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))) ) ).

% power_less_power_Suc
tff(fact_4052_power__less__power__Suc,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na))) ) ).

% power_less_power_Suc
tff(fact_4053_power__less__power__Suc,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))) ) ).

% power_less_power_Suc
tff(fact_4054_power__less__power__Suc,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))) ) ).

% power_less_power_Suc
tff(fact_4055_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_literal,C3: nat] :
      ( ! [G2: set_literal] :
          ( ord_le7307670543136651348iteral(G2,F2)
         => ( finite5847741373460823677iteral(G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(G2)),C3) ) )
     => ( finite5847741373460823677iteral(F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(F2)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_4056_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_list_nat,C3: nat] :
      ( ! [G2: set_list_nat] :
          ( ord_le6045566169113846134st_nat(G2,F2)
         => ( finite8100373058378681591st_nat(G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(G2)),C3) ) )
     => ( finite8100373058378681591st_nat(F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(F2)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_4057_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_set_nat,C3: nat] :
      ( ! [G2: set_set_nat] :
          ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(G2),F2)
         => ( aa_set_set_nat_o(finite1152437895449049373et_nat,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(G2)),C3) ) )
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(F2)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_4058_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_nat,C3: nat] :
      ( ! [G2: set_nat] :
          ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,G2),F2)
         => ( aa_set_nat_o(finite_finite_nat,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,G2)),C3) ) )
     => ( aa_set_nat_o(finite_finite_nat,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,F2)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_4059_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_complex,C3: nat] :
      ( ! [G2: set_complex] :
          ( aa_set_complex_o(ord_le211207098394363844omplex(G2),F2)
         => ( aa_set_complex_o(finite3207457112153483333omplex,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(G2)),C3) ) )
     => ( aa_set_complex_o(finite3207457112153483333omplex,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(F2)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_4060_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Code_integer,C3: nat] :
      ( ! [G2: set_Code_integer] :
          ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(G2),F2)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(G2)),C3) ) )
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(F2)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_4061_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_int,C3: nat] :
      ( ! [G2: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(G2),F2)
         => ( aa_set_int_o(finite_finite_int,G2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(G2)),C3) ) )
     => ( aa_set_int_o(finite_finite_int,F2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(F2)),C3) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_4062_card__seteq,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( finite5847741373460823677iteral(B4)
     => ( ord_le7307670543136651348iteral(A4,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(B4)),finite_card_literal(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_4063_card__seteq,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le6045566169113846134st_nat(A4,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(B4)),finite_card_list_nat(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_4064_card__seteq,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(B4)),finite_card_set_nat(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_4065_card__seteq,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B4)),aa_set_nat_nat(finite_card_nat,A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_4066_card__seteq,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(B4)),finite_card_complex(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_4067_card__seteq,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(B4)),finite4902975817058060853nteger(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_4068_card__seteq,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(B4)),finite_card_int(A4))
         => ( A4 = B4 ) ) ) ) ).

% card_seteq
tff(fact_4069_card__mono,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( finite5847741373460823677iteral(B4)
     => ( ord_le7307670543136651348iteral(A4,B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(A4)),finite_card_literal(B4)) ) ) ).

% card_mono
tff(fact_4070_card__mono,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le6045566169113846134st_nat(A4,B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4)) ) ) ).

% card_mono
tff(fact_4071_card__mono,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4)) ) ) ).

% card_mono
tff(fact_4072_card__mono,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4)) ) ) ).

% card_mono
tff(fact_4073_card__mono,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A4)),finite_card_complex(B4)) ) ) ).

% card_mono
tff(fact_4074_card__mono,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4)) ) ) ).

% card_mono
tff(fact_4075_card__mono,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B4)) ) ) ).

% card_mono
tff(fact_4076_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S2: set_literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_literal(S2))
     => ~ ! [T4: set_literal] :
            ( ord_le7307670543136651348iteral(T4,S2)
           => ( ( finite_card_literal(T4) = Na )
             => ~ finite5847741373460823677iteral(T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4077_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S2: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_list_nat(S2))
     => ~ ! [T4: set_list_nat] :
            ( ord_le6045566169113846134st_nat(T4,S2)
           => ( ( finite_card_list_nat(T4) = Na )
             => ~ finite8100373058378681591st_nat(T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4078_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S2: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_set_nat(S2))
     => ~ ! [T4: set_set_nat] :
            ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(T4),S2)
           => ( ( finite_card_set_nat(T4) = Na )
             => ~ aa_set_set_nat_o(finite1152437895449049373et_nat,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4079_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_set_nat_nat(finite_card_nat,S2))
     => ~ ! [T4: set_nat] :
            ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,T4),S2)
           => ( ( aa_set_nat_nat(finite_card_nat,T4) = Na )
             => ~ aa_set_nat_o(finite_finite_nat,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4080_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S2: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_complex(S2))
     => ~ ! [T4: set_complex] :
            ( aa_set_complex_o(ord_le211207098394363844omplex(T4),S2)
           => ( ( finite_card_complex(T4) = Na )
             => ~ aa_set_complex_o(finite3207457112153483333omplex,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4081_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S2: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite4902975817058060853nteger(S2))
     => ~ ! [T4: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(T4),S2)
           => ( ( finite4902975817058060853nteger(T4) = Na )
             => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4082_obtain__subset__with__card__n,axiom,
    ! [Na: nat,S2: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_int(S2))
     => ~ ! [T4: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(T4),S2)
           => ( ( finite_card_int(T4) = Na )
             => ~ aa_set_int_o(finite_finite_int,T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4083_power__gt1,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_4084_power__gt1,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_4085_power__gt1,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_4086_power__gt1,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_4087_power__0__left,axiom,
    ! [Na: nat] :
      ( aa_nat_uint32(power_power_uint32(zero_zero_uint32),Na) = $ite(Na = zero_zero_nat,one_one_uint32,zero_zero_uint32) ) ).

% power_0_left
tff(fact_4088_power__0__left,axiom,
    ! [Na: nat] :
      ( aa_nat_rat(power_power_rat(zero_zero_rat),Na) = $ite(Na = zero_zero_nat,one_one_rat,zero_zero_rat) ) ).

% power_0_left
tff(fact_4089_power__0__left,axiom,
    ! [Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = $ite(Na = zero_zero_nat,one_one_nat,zero_zero_nat) ) ).

% power_0_left
tff(fact_4090_power__0__left,axiom,
    ! [Na: nat] :
      ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = $ite(Na = zero_zero_nat,one_one_real,zero_zero_real) ) ).

% power_0_left
tff(fact_4091_power__0__left,axiom,
    ! [Na: nat] :
      ( aa_nat_complex(power_power_complex(zero_zero_complex),Na) = $ite(Na = zero_zero_nat,one_one_complex,zero_zero_complex) ) ).

% power_0_left
tff(fact_4092_power__0__left,axiom,
    ! [Na: nat] :
      ( aa_nat_int(power_power_int(zero_zero_int),Na) = $ite(Na = zero_zero_nat,one_one_int,zero_zero_int) ) ).

% power_0_left
tff(fact_4093_card__1__singletonE,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( finite7802652506058667612T_VEBT(A4) = one_one_nat )
     => ~ ! [X: vEBT_VEBT] : ( A4 != aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X),bot_bo8194388402131092736T_VEBT) ) ) ).

% card_1_singletonE
tff(fact_4094_card__1__singletonE,axiom,
    ! [A4: set_complex] :
      ( ( finite_card_complex(A4) = one_one_nat )
     => ~ ! [X: complex] : ( A4 != insert_complex(X,bot_bot_set_complex) ) ) ).

% card_1_singletonE
tff(fact_4095_card__1__singletonE,axiom,
    ! [A4: set_literal] :
      ( ( finite_card_literal(A4) = one_one_nat )
     => ~ ! [X: literal] : ( A4 != insert_literal(X,bot_bot_set_literal) ) ) ).

% card_1_singletonE
tff(fact_4096_card__1__singletonE,axiom,
    ! [A4: set_list_nat] :
      ( ( finite_card_list_nat(A4) = one_one_nat )
     => ~ ! [X: list_nat] : ( A4 != insert_list_nat(X,bot_bot_set_list_nat) ) ) ).

% card_1_singletonE
tff(fact_4097_card__1__singletonE,axiom,
    ! [A4: set_set_nat] :
      ( ( finite_card_set_nat(A4) = one_one_nat )
     => ~ ! [X: set_nat] : ( A4 != aa_set1013257548857649334et_nat(insert_set_nat(X),bot_bot_set_set_nat) ) ) ).

% card_1_singletonE
tff(fact_4098_card__1__singletonE,axiom,
    ! [A4: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = one_one_nat )
     => ~ ! [X: nat] : ( A4 != aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat) ) ) ).

% card_1_singletonE
tff(fact_4099_card__1__singletonE,axiom,
    ! [A4: set_int] :
      ( ( finite_card_int(A4) = one_one_nat )
     => ~ ! [X: int] : ( A4 != aa_set_int_set_int(insert_int(X),bot_bot_set_int) ) ) ).

% card_1_singletonE
tff(fact_4100_card__1__singletonE,axiom,
    ! [A4: set_o] :
      ( ( finite_card_o(A4) = one_one_nat )
     => ~ ! [X: $o] : ( A4 != aa_set_o_set_o(insert_o((X)),bot_bot_set_o) ) ) ).

% card_1_singletonE
tff(fact_4101_power__strict__increasing,axiom,
    ! [Na: nat,N6: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N6)) ) ) ).

% power_strict_increasing
tff(fact_4102_power__strict__increasing,axiom,
    ! [Na: nat,N6: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N6)) ) ) ).

% power_strict_increasing
tff(fact_4103_power__strict__increasing,axiom,
    ! [Na: nat,N6: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N6)) ) ) ).

% power_strict_increasing
tff(fact_4104_power__strict__increasing,axiom,
    ! [Na: nat,N6: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N6)) ) ) ).

% power_strict_increasing
tff(fact_4105_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_4106_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_4107_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_4108_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_4109_card__less__sym__Diff,axiom,
    ! [A4: set_literal,B4: set_literal] :
      ( finite5847741373460823677iteral(A4)
     => ( finite5847741373460823677iteral(B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_literal(A4)),finite_card_literal(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_literal(minus_7832829386415567259iteral(A4,B4))),finite_card_literal(minus_7832829386415567259iteral(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4110_card__less__sym__Diff,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite8100373058378681591st_nat(B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,B4))),finite_card_list_nat(minus_7954133019191499631st_nat(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4111_card__less__sym__Diff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,B4))),finite_card_set_nat(minus_2163939370556025621et_nat(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4112_card__less__sym__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A4)),finite_card_int(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(A4,B4))),finite_card_int(minus_minus_set_int(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4113_card__less__sym__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(A4)),finite_card_complex(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A4,B4))),finite_card_complex(minus_811609699411566653omplex(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4114_card__less__sym__Diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,B4))),finite4902975817058060853nteger(minus_2355218937544613996nteger(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4115_card__less__sym__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,B4))),aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(B4,A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4116_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_uint32(power_power_uint32(zero_zero_uint32),Na) = zero_zero_uint32 ) ) ).

% zero_power
tff(fact_4117_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Na) = zero_zero_rat ) ) ).

% zero_power
tff(fact_4118_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = zero_zero_nat ) ) ).

% zero_power
tff(fact_4119_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = zero_zero_real ) ) ).

% zero_power
tff(fact_4120_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Na) = zero_zero_complex ) ) ).

% zero_power
tff(fact_4121_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(power_power_int(zero_zero_int),Na) = zero_zero_int ) ) ).

% zero_power
tff(fact_4122_power__increasing,axiom,
    ! [Na: nat,N6: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N6)) ) ) ).

% power_increasing
tff(fact_4123_power__increasing,axiom,
    ! [Na: nat,N6: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(A),N6)) ) ) ).

% power_increasing
tff(fact_4124_power__increasing,axiom,
    ! [Na: nat,N6: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N6)) ) ) ).

% power_increasing
tff(fact_4125_power__increasing,axiom,
    ! [Na: nat,N6: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),N6)) ) ) ).

% power_increasing
tff(fact_4126_card__le__sym__Diff,axiom,
    ! [A4: set_literal,B4: set_literal] :
      ( finite5847741373460823677iteral(A4)
     => ( finite5847741373460823677iteral(B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(A4)),finite_card_literal(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(minus_7832829386415567259iteral(A4,B4))),finite_card_literal(minus_7832829386415567259iteral(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4127_card__le__sym__Diff,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite8100373058378681591st_nat(B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,B4))),finite_card_list_nat(minus_7954133019191499631st_nat(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4128_card__le__sym__Diff,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,B4))),finite_card_set_nat(minus_2163939370556025621et_nat(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4129_card__le__sym__Diff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(minus_minus_set_int(A4,B4))),finite_card_int(minus_minus_set_int(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4130_card__le__sym__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A4)),finite_card_complex(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(minus_811609699411566653omplex(A4,B4))),finite_card_complex(minus_811609699411566653omplex(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4131_card__le__sym__Diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,B4))),finite4902975817058060853nteger(minus_2355218937544613996nteger(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4132_card__le__sym__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,B4))),aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(B4,A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4133_psubset__card__mono,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( finite5847741373460823677iteral(B4)
     => ( ord_less_set_literal(A4,B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_literal(A4)),finite_card_literal(B4)) ) ) ).

% psubset_card_mono
tff(fact_4134_psubset__card__mono,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le1190675801316882794st_nat(A4,B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4)) ) ) ).

% psubset_card_mono
tff(fact_4135_psubset__card__mono,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_less_set_set_nat(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4)) ) ) ).

% psubset_card_mono
tff(fact_4136_psubset__card__mono,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4)) ) ) ).

% psubset_card_mono
tff(fact_4137_psubset__card__mono,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_set_int(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A4)),finite_card_int(B4)) ) ) ).

% psubset_card_mono
tff(fact_4138_psubset__card__mono,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_less_set_complex(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(A4)),finite_card_complex(B4)) ) ) ).

% psubset_card_mono
tff(fact_4139_psubset__card__mono,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(A4),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4)) ) ) ).

% psubset_card_mono
tff(fact_4140_length__pos__if__in__set,axiom,
    ! [Xa: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_4141_length__pos__if__in__set,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_4142_length__pos__if__in__set,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_4143_length__pos__if__in__set,axiom,
    ! [Xa: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_4144_length__pos__if__in__set,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_4145_length__pos__if__in__set,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_nat_nat(size_size_list_nat,Xs)) ) ).

% length_pos_if_in_set
tff(fact_4146_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),K)) ) ).

% power_gt_expt
tff(fact_4147_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na)) ) ).

% nat_one_le_power
tff(fact_4148_finite__enum__ext,axiom,
    ! [X4: set_nat,Y6: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_set_nat_nat(finite_card_nat,X4))
         => ( aa_nat_nat(infini8530281810654367211te_nat(X4),I2) = aa_nat_nat(infini8530281810654367211te_nat(Y6),I2) ) )
     => ( aa_set_nat_o(finite_finite_nat,X4)
       => ( aa_set_nat_o(finite_finite_nat,Y6)
         => ( ( aa_set_nat_nat(finite_card_nat,X4) = aa_set_nat_nat(finite_card_nat,Y6) )
           => ( X4 = Y6 ) ) ) ) ) ).

% finite_enum_ext
tff(fact_4149_finite__enumerate__Ex,axiom,
    ! [S2: set_nat,Sa: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(member_nat(Sa),S2)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_set_nat_nat(finite_card_nat,S2))
            & ( aa_nat_nat(infini8530281810654367211te_nat(S2),N) = Sa ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_4150_finite__enumerate__in__set,axiom,
    ! [S2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_set_nat_nat(finite_card_nat,S2))
       => aa_set_nat_o(member_nat(aa_nat_nat(infini8530281810654367211te_nat(S2),Na)),S2) ) ) ).

% finite_enumerate_in_set
tff(fact_4151_length__removeAll__less,axiom,
    ! [Xa: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(removeAll_int(Xa,Xs))),size_size_list_int(Xs)) ) ).

% length_removeAll_less
tff(fact_4152_length__removeAll__less,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s3254054031482475050et_nat(removeAll_set_nat(Xa,Xs))),size_s3254054031482475050et_nat(Xs)) ) ).

% length_removeAll_less
tff(fact_4153_length__removeAll__less,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(removeAll_VEBT_VEBT(Xa,Xs))),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_removeAll_less
tff(fact_4154_length__removeAll__less,axiom,
    ! [Xa: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(removeAll_real(Xa,Xs))),size_size_list_real(Xs)) ) ).

% length_removeAll_less
tff(fact_4155_length__removeAll__less,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(removeAll_o((Xa),Xs))),size_size_list_o(Xs)) ) ).

% length_removeAll_less
tff(fact_4156_length__removeAll__less,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,removeAll_nat(Xa,Xs))),aa_list_nat_nat(size_size_list_nat,Xs)) ) ).

% length_removeAll_less
tff(fact_4157_card__gt__0__iff,axiom,
    ! [A4: set_literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_literal(A4))
    <=> ( ( A4 != bot_bot_set_literal )
        & finite5847741373460823677iteral(A4) ) ) ).

% card_gt_0_iff
tff(fact_4158_card__gt__0__iff,axiom,
    ! [A4: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_list_nat(A4))
    <=> ( ( A4 != bot_bot_set_list_nat )
        & finite8100373058378681591st_nat(A4) ) ) ).

% card_gt_0_iff
tff(fact_4159_card__gt__0__iff,axiom,
    ! [A4: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_set_nat(A4))
    <=> ( ( A4 != bot_bot_set_set_nat )
        & aa_set_set_nat_o(finite1152437895449049373et_nat,A4) ) ) ).

% card_gt_0_iff
tff(fact_4160_card__gt__0__iff,axiom,
    ! [A4: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_complex(A4))
    <=> ( ( A4 != bot_bot_set_complex )
        & aa_set_complex_o(finite3207457112153483333omplex,A4) ) ) ).

% card_gt_0_iff
tff(fact_4161_card__gt__0__iff,axiom,
    ! [A4: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite4902975817058060853nteger(A4))
    <=> ( ( A4 != bot_bo3990330152332043303nteger )
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ) ).

% card_gt_0_iff
tff(fact_4162_card__gt__0__iff,axiom,
    ! [A4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A4))
    <=> ( ( A4 != bot_bot_set_nat )
        & aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% card_gt_0_iff
tff(fact_4163_card__gt__0__iff,axiom,
    ! [A4: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A4))
    <=> ( ( A4 != bot_bot_set_int )
        & aa_set_int_o(finite_finite_int,A4) ) ) ).

% card_gt_0_iff
tff(fact_4164_card__gt__0__iff,axiom,
    ! [A4: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_o(A4))
    <=> ( ( A4 != bot_bot_set_o )
        & aa_set_o_o(finite_finite_o,A4) ) ) ).

% card_gt_0_iff
tff(fact_4165_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_literal] :
      ( finite5847741373460823677iteral(A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(A4)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X5: literal] :
            ( member_literal(X5,A4)
           => ! [Xa3: literal] :
                ( member_literal(Xa3,A4)
               => ( X5 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4166_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(A4)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X5: list_nat] :
            ( member_list_nat(X5,A4)
           => ! [Xa3: list_nat] :
                ( member_list_nat(Xa3,A4)
               => ( X5 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4167_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(A4)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X5: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X5),A4)
           => ! [Xa3: set_nat] :
                ( aa_set_set_nat_o(member_set_nat(Xa3),A4)
               => ( X5 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4168_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
           => ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A4)
               => ( X5 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4169_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
           => ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A4)
               => ( X5 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4170_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(A4)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
           => ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A4)
               => ( X5 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4171_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite4902975817058060853nteger(A4)),aa_nat_nat(suc,zero_zero_nat))
      <=> ! [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
           => ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A4)
               => ( X5 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4172_card__Suc__eq,axiom,
    ! [A4: set_VEBT_VEBT,K: nat] :
      ( ( finite7802652506058667612T_VEBT(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: vEBT_VEBT,B8: set_VEBT_VEBT] :
          ( ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B6),B8) )
          & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B6),B8)
          & ( finite7802652506058667612T_VEBT(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% card_Suc_eq
tff(fact_4173_card__Suc__eq,axiom,
    ! [A4: set_real,K: nat] :
      ( ( finite_card_real(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: real,B8: set_real] :
          ( ( A4 = aa_set_real_set_real(insert_real(B6),B8) )
          & ~ aa_set_real_o(member_real(B6),B8)
          & ( finite_card_real(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_real ) ) ) ) ).

% card_Suc_eq
tff(fact_4174_card__Suc__eq,axiom,
    ! [A4: set_complex,K: nat] :
      ( ( finite_card_complex(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: complex,B8: set_complex] :
          ( ( A4 = insert_complex(B6,B8) )
          & ~ aa_set_complex_o(member_complex(B6),B8)
          & ( finite_card_complex(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_complex ) ) ) ) ).

% card_Suc_eq
tff(fact_4175_card__Suc__eq,axiom,
    ! [A4: set_literal,K: nat] :
      ( ( finite_card_literal(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: literal,B8: set_literal] :
          ( ( A4 = insert_literal(B6,B8) )
          & ~ member_literal(B6,B8)
          & ( finite_card_literal(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_literal ) ) ) ) ).

% card_Suc_eq
tff(fact_4176_card__Suc__eq,axiom,
    ! [A4: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: list_nat,B8: set_list_nat] :
          ( ( A4 = insert_list_nat(B6,B8) )
          & ~ member_list_nat(B6,B8)
          & ( finite_card_list_nat(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_list_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_4177_card__Suc__eq,axiom,
    ! [A4: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: set_nat,B8: set_set_nat] :
          ( ( A4 = aa_set1013257548857649334et_nat(insert_set_nat(B6),B8) )
          & ~ aa_set_set_nat_o(member_set_nat(B6),B8)
          & ( finite_card_set_nat(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_4178_card__Suc__eq,axiom,
    ! [A4: set_nat,K: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: nat,B8: set_nat] :
          ( ( A4 = aa_set_nat_set_nat(insert_nat(B6),B8) )
          & ~ aa_set_nat_o(member_nat(B6),B8)
          & ( aa_set_nat_nat(finite_card_nat,B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_nat ) ) ) ) ).

% card_Suc_eq
tff(fact_4179_card__Suc__eq,axiom,
    ! [A4: set_int,K: nat] :
      ( ( finite_card_int(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: int,B8: set_int] :
          ( ( A4 = aa_set_int_set_int(insert_int(B6),B8) )
          & ~ aa_set_int_o(member_int(B6),B8)
          & ( finite_card_int(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_int ) ) ) ) ).

% card_Suc_eq
tff(fact_4180_card__Suc__eq,axiom,
    ! [A4: set_o,K: nat] :
      ( ( finite_card_o(A4) = aa_nat_nat(suc,K) )
    <=> ? [B6: $o,B8: set_o] :
          ( ( A4 = aa_set_o_set_o(insert_o((B6)),B8) )
          & ~ aa_set_o_o(member_o((B6)),B8)
          & ( finite_card_o(B8) = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_o ) ) ) ) ).

% card_Suc_eq
tff(fact_4181_card__eq__SucD,axiom,
    ! [A4: set_VEBT_VEBT,K: nat] :
      ( ( finite7802652506058667612T_VEBT(A4) = aa_nat_nat(suc,K) )
     => ? [B3: vEBT_VEBT,B7: set_VEBT_VEBT] :
          ( ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(B3),B7) )
          & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),B7)
          & ( finite7802652506058667612T_VEBT(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% card_eq_SucD
tff(fact_4182_card__eq__SucD,axiom,
    ! [A4: set_real,K: nat] :
      ( ( finite_card_real(A4) = aa_nat_nat(suc,K) )
     => ? [B3: real,B7: set_real] :
          ( ( A4 = aa_set_real_set_real(insert_real(B3),B7) )
          & ~ aa_set_real_o(member_real(B3),B7)
          & ( finite_card_real(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_real ) ) ) ) ).

% card_eq_SucD
tff(fact_4183_card__eq__SucD,axiom,
    ! [A4: set_complex,K: nat] :
      ( ( finite_card_complex(A4) = aa_nat_nat(suc,K) )
     => ? [B3: complex,B7: set_complex] :
          ( ( A4 = insert_complex(B3,B7) )
          & ~ aa_set_complex_o(member_complex(B3),B7)
          & ( finite_card_complex(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_complex ) ) ) ) ).

% card_eq_SucD
tff(fact_4184_card__eq__SucD,axiom,
    ! [A4: set_literal,K: nat] :
      ( ( finite_card_literal(A4) = aa_nat_nat(suc,K) )
     => ? [B3: literal,B7: set_literal] :
          ( ( A4 = insert_literal(B3,B7) )
          & ~ member_literal(B3,B7)
          & ( finite_card_literal(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_literal ) ) ) ) ).

% card_eq_SucD
tff(fact_4185_card__eq__SucD,axiom,
    ! [A4: set_list_nat,K: nat] :
      ( ( finite_card_list_nat(A4) = aa_nat_nat(suc,K) )
     => ? [B3: list_nat,B7: set_list_nat] :
          ( ( A4 = insert_list_nat(B3,B7) )
          & ~ member_list_nat(B3,B7)
          & ( finite_card_list_nat(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_list_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_4186_card__eq__SucD,axiom,
    ! [A4: set_set_nat,K: nat] :
      ( ( finite_card_set_nat(A4) = aa_nat_nat(suc,K) )
     => ? [B3: set_nat,B7: set_set_nat] :
          ( ( A4 = aa_set1013257548857649334et_nat(insert_set_nat(B3),B7) )
          & ~ aa_set_set_nat_o(member_set_nat(B3),B7)
          & ( finite_card_set_nat(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_4187_card__eq__SucD,axiom,
    ! [A4: set_nat,K: nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,K) )
     => ? [B3: nat,B7: set_nat] :
          ( ( A4 = aa_set_nat_set_nat(insert_nat(B3),B7) )
          & ~ aa_set_nat_o(member_nat(B3),B7)
          & ( aa_set_nat_nat(finite_card_nat,B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
tff(fact_4188_card__eq__SucD,axiom,
    ! [A4: set_int,K: nat] :
      ( ( finite_card_int(A4) = aa_nat_nat(suc,K) )
     => ? [B3: int,B7: set_int] :
          ( ( A4 = aa_set_int_set_int(insert_int(B3),B7) )
          & ~ aa_set_int_o(member_int(B3),B7)
          & ( finite_card_int(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
tff(fact_4189_card__eq__SucD,axiom,
    ! [A4: set_o,K: nat] :
      ( ( finite_card_o(A4) = aa_nat_nat(suc,K) )
     => ? [B3: $o,B7: set_o] :
          ( ( A4 = aa_set_o_set_o(insert_o((B3)),B7) )
          & ~ aa_set_o_o(member_o((B3)),B7)
          & ( finite_card_o(B7) = K )
          & ( ( K = zero_zero_nat )
           => ( B7 = bot_bot_set_o ) ) ) ) ).

% card_eq_SucD
tff(fact_4190_card__1__singleton__iff,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( finite7802652506058667612T_VEBT(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: vEBT_VEBT] : ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(X5),bot_bo8194388402131092736T_VEBT) ) ) ).

% card_1_singleton_iff
tff(fact_4191_card__1__singleton__iff,axiom,
    ! [A4: set_complex] :
      ( ( finite_card_complex(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: complex] : ( A4 = insert_complex(X5,bot_bot_set_complex) ) ) ).

% card_1_singleton_iff
tff(fact_4192_card__1__singleton__iff,axiom,
    ! [A4: set_literal] :
      ( ( finite_card_literal(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: literal] : ( A4 = insert_literal(X5,bot_bot_set_literal) ) ) ).

% card_1_singleton_iff
tff(fact_4193_card__1__singleton__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( finite_card_list_nat(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: list_nat] : ( A4 = insert_list_nat(X5,bot_bot_set_list_nat) ) ) ).

% card_1_singleton_iff
tff(fact_4194_card__1__singleton__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( finite_card_set_nat(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: set_nat] : ( A4 = aa_set1013257548857649334et_nat(insert_set_nat(X5),bot_bot_set_set_nat) ) ) ).

% card_1_singleton_iff
tff(fact_4195_card__1__singleton__iff,axiom,
    ! [A4: set_nat] :
      ( ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: nat] : ( A4 = aa_set_nat_set_nat(insert_nat(X5),bot_bot_set_nat) ) ) ).

% card_1_singleton_iff
tff(fact_4196_card__1__singleton__iff,axiom,
    ! [A4: set_int] :
      ( ( finite_card_int(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: int] : ( A4 = aa_set_int_set_int(insert_int(X5),bot_bot_set_int) ) ) ).

% card_1_singleton_iff
tff(fact_4197_card__1__singleton__iff,axiom,
    ! [A4: set_o] :
      ( ( finite_card_o(A4) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ? [X5: $o] : ( A4 = aa_set_o_set_o(insert_o((X5)),bot_bot_set_o) ) ) ).

% card_1_singleton_iff
tff(fact_4198_power__Suc__less,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% power_Suc_less
tff(fact_4199_power__Suc__less,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Na))),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% power_Suc_less
tff(fact_4200_power__Suc__less,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% power_Suc_less
tff(fact_4201_power__Suc__less,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Na))),aa_nat_int(power_power_int(A),Na)) ) ) ).

% power_Suc_less
tff(fact_4202_power__Suc__le__self,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_4203_power__Suc__le__self,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_4204_power__Suc__le__self,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_4205_power__Suc__le__self,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),A) ) ) ).

% power_Suc_le_self
tff(fact_4206_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_o(A4))
    <=> ? [A6: $o,B8: set_o] :
          ( ( A4 = aa_set_o_set_o(insert_o((A6)),B8) )
          & ~ aa_set_o_o(member_o((A6)),B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_o(B8))
          & aa_set_o_o(finite_finite_o,B8) ) ) ).

% card_le_Suc_iff
tff(fact_4207_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite7802652506058667612T_VEBT(A4))
    <=> ? [A6: vEBT_VEBT,B8: set_VEBT_VEBT] :
          ( ( A4 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A6),B8) )
          & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A6),B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite7802652506058667612T_VEBT(B8))
          & aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B8) ) ) ).

% card_le_Suc_iff
tff(fact_4208_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_real(A4))
    <=> ? [A6: real,B8: set_real] :
          ( ( A4 = aa_set_real_set_real(insert_real(A6),B8) )
          & ~ aa_set_real_o(member_real(A6),B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_real(B8))
          & aa_set_real_o(finite_finite_real,B8) ) ) ).

% card_le_Suc_iff
tff(fact_4209_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_literal(A4))
    <=> ? [A6: literal,B8: set_literal] :
          ( ( A4 = insert_literal(A6,B8) )
          & ~ member_literal(A6,B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_literal(B8))
          & finite5847741373460823677iteral(B8) ) ) ).

% card_le_Suc_iff
tff(fact_4210_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_list_nat(A4))
    <=> ? [A6: list_nat,B8: set_list_nat] :
          ( ( A4 = insert_list_nat(A6,B8) )
          & ~ member_list_nat(A6,B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_list_nat(B8))
          & finite8100373058378681591st_nat(B8) ) ) ).

% card_le_Suc_iff
tff(fact_4211_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_set_nat(A4))
    <=> ? [A6: set_nat,B8: set_set_nat] :
          ( ( A4 = aa_set1013257548857649334et_nat(insert_set_nat(A6),B8) )
          & ~ aa_set_set_nat_o(member_set_nat(A6),B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_set_nat(B8))
          & aa_set_set_nat_o(finite1152437895449049373et_nat,B8) ) ) ).

% card_le_Suc_iff
tff(fact_4212_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_set_nat_nat(finite_card_nat,A4))
    <=> ? [A6: nat,B8: set_nat] :
          ( ( A4 = aa_set_nat_set_nat(insert_nat(A6),B8) )
          & ~ aa_set_nat_o(member_nat(A6),B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_set_nat_nat(finite_card_nat,B8))
          & aa_set_nat_o(finite_finite_nat,B8) ) ) ).

% card_le_Suc_iff
tff(fact_4213_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_int(A4))
    <=> ? [A6: int,B8: set_int] :
          ( ( A4 = aa_set_int_set_int(insert_int(A6),B8) )
          & ~ aa_set_int_o(member_int(A6),B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_int(B8))
          & aa_set_int_o(finite_finite_int,B8) ) ) ).

% card_le_Suc_iff
tff(fact_4214_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite_card_complex(A4))
    <=> ? [A6: complex,B8: set_complex] :
          ( ( A4 = insert_complex(A6,B8) )
          & ~ aa_set_complex_o(member_complex(A6),B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite_card_complex(B8))
          & aa_set_complex_o(finite3207457112153483333omplex,B8) ) ) ).

% card_le_Suc_iff
tff(fact_4215_card__le__Suc__iff,axiom,
    ! [Na: nat,A4: set_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),finite4902975817058060853nteger(A4))
    <=> ? [A6: code_integer,B8: set_Code_integer] :
          ( ( A4 = insert_Code_integer(A6,B8) )
          & ~ aa_set5643387332196575111eger_o(member_Code_integer(A6),B8)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),finite4902975817058060853nteger(B8))
          & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B8) ) ) ).

% card_le_Suc_iff
tff(fact_4216_power__Suc__less__one,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Na))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_4217_power__Suc__less__one,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Na))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_4218_power__Suc__less__one,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Na))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_4219_power__Suc__less__one,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,Na))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_4220_surj__card__le,axiom,
    ! [A4: set_literal,B4: set_complex,F: fun_literal_complex] :
      ( finite5847741373460823677iteral(A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_5274195009022015549omplex(F,A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(B4)),finite_card_literal(A4)) ) ) ).

% surj_card_le
tff(fact_4221_surj__card__le,axiom,
    ! [A4: set_literal,B4: set_literal,F: fun_literal_literal] :
      ( finite5847741373460823677iteral(A4)
     => ( ord_le7307670543136651348iteral(B4,image_8195128725298311301iteral(F,A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(B4)),finite_card_literal(A4)) ) ) ).

% surj_card_le
tff(fact_4222_surj__card__le,axiom,
    ! [A4: set_literal,B4: set_nat,F: fun_literal_nat] :
      ( finite5847741373460823677iteral(A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),image_literal_nat(F,A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B4)),finite_card_literal(A4)) ) ) ).

% surj_card_le
tff(fact_4223_surj__card__le,axiom,
    ! [A4: set_nat,B4: set_complex,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_nat_complex(F,A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(B4)),aa_set_nat_nat(finite_card_nat,A4)) ) ) ).

% surj_card_le
tff(fact_4224_surj__card__le,axiom,
    ! [A4: set_nat,B4: set_literal,F: fun_nat_literal] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ord_le7307670543136651348iteral(B4,image_nat_literal(F,A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(B4)),aa_set_nat_nat(finite_card_nat,A4)) ) ) ).

% surj_card_le
tff(fact_4225_surj__card__le,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B4)),aa_set_nat_nat(finite_card_nat,A4)) ) ) ).

% surj_card_le
tff(fact_4226_surj__card__le,axiom,
    ! [A4: set_int,B4: set_complex,F: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_int_complex(F,A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(B4)),finite_card_int(A4)) ) ) ).

% surj_card_le
tff(fact_4227_surj__card__le,axiom,
    ! [A4: set_int,B4: set_literal,F: fun_int_literal] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ord_le7307670543136651348iteral(B4,image_int_literal(F,A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(B4)),finite_card_int(A4)) ) ) ).

% surj_card_le
tff(fact_4228_surj__card__le,axiom,
    ! [A4: set_int,B4: set_nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),aa_set_int_set_nat(image_int_nat(F),A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B4)),finite_card_int(A4)) ) ) ).

% surj_card_le
tff(fact_4229_surj__card__le,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),image_1468599708987790691omplex(F,A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(B4)),finite_card_complex(A4)) ) ) ).

% surj_card_le
tff(fact_4230_power__strict__decreasing,axiom,
    ! [Na: nat,N6: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N6)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_4231_power__strict__decreasing,axiom,
    ! [Na: nat,N6: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N6)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_4232_power__strict__decreasing,axiom,
    ! [Na: nat,N6: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N6)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_4233_power__strict__decreasing,axiom,
    ! [Na: nat,N6: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N6)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_4234_card__1__singletonI,axiom,
    ! [S2: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( ( finite7802652506058667612T_VEBT(S2) = one_one_nat )
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),S2)
         => ( S2 = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT) ) ) ) ) ).

% card_1_singletonI
tff(fact_4235_card__1__singletonI,axiom,
    ! [S2: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( ( finite_card_real(S2) = one_one_nat )
       => ( aa_set_real_o(member_real(Xa),S2)
         => ( S2 = aa_set_real_set_real(insert_real(Xa),bot_bot_set_real) ) ) ) ) ).

% card_1_singletonI
tff(fact_4236_card__1__singletonI,axiom,
    ! [S2: set_literal,Xa: literal] :
      ( finite5847741373460823677iteral(S2)
     => ( ( finite_card_literal(S2) = one_one_nat )
       => ( member_literal(Xa,S2)
         => ( S2 = insert_literal(Xa,bot_bot_set_literal) ) ) ) ) ).

% card_1_singletonI
tff(fact_4237_card__1__singletonI,axiom,
    ! [S2: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(S2)
     => ( ( finite_card_list_nat(S2) = one_one_nat )
       => ( member_list_nat(Xa,S2)
         => ( S2 = insert_list_nat(Xa,bot_bot_set_list_nat) ) ) ) ) ).

% card_1_singletonI
tff(fact_4238_card__1__singletonI,axiom,
    ! [S2: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,S2)
     => ( ( finite_card_set_nat(S2) = one_one_nat )
       => ( aa_set_set_nat_o(member_set_nat(Xa),S2)
         => ( S2 = aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat) ) ) ) ) ).

% card_1_singletonI
tff(fact_4239_card__1__singletonI,axiom,
    ! [S2: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( finite_card_complex(S2) = one_one_nat )
       => ( aa_set_complex_o(member_complex(Xa),S2)
         => ( S2 = insert_complex(Xa,bot_bot_set_complex) ) ) ) ) ).

% card_1_singletonI
tff(fact_4240_card__1__singletonI,axiom,
    ! [S2: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( finite4902975817058060853nteger(S2) = one_one_nat )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),S2)
         => ( S2 = insert_Code_integer(Xa,bot_bo3990330152332043303nteger) ) ) ) ) ).

% card_1_singletonI
tff(fact_4241_card__1__singletonI,axiom,
    ! [S2: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( aa_set_nat_nat(finite_card_nat,S2) = one_one_nat )
       => ( aa_set_nat_o(member_nat(Xa),S2)
         => ( S2 = aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat) ) ) ) ) ).

% card_1_singletonI
tff(fact_4242_card__1__singletonI,axiom,
    ! [S2: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( finite_card_int(S2) = one_one_nat )
       => ( aa_set_int_o(member_int(Xa),S2)
         => ( S2 = aa_set_int_set_int(insert_int(Xa),bot_bot_set_int) ) ) ) ) ).

% card_1_singletonI
tff(fact_4243_card__1__singletonI,axiom,
    ! [S2: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,S2)
     => ( ( finite_card_o(S2) = one_one_nat )
       => ( aa_set_o_o(member_o((Xa)),S2)
         => ( S2 = aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o) ) ) ) ) ).

% card_1_singletonI
tff(fact_4244_power__decreasing,axiom,
    ! [Na: nat,N6: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N6)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ) ).

% power_decreasing
tff(fact_4245_power__decreasing,axiom,
    ! [Na: nat,N6: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N6)),aa_nat_rat(power_power_rat(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_4246_power__decreasing,axiom,
    ! [Na: nat,N6: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N6)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_decreasing
tff(fact_4247_power__decreasing,axiom,
    ! [Na: nat,N6: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N6)),aa_nat_int(power_power_int(A),Na)) ) ) ) ).

% power_decreasing
tff(fact_4248_power__eq__imp__eq__base,axiom,
    ! [A: real,Na: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_4249_power__eq__imp__eq__base,axiom,
    ! [A: rat,Na: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B),Na) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_4250_power__eq__imp__eq__base,axiom,
    ! [A: nat,Na: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_4251_power__eq__imp__eq__base,axiom,
    ! [A: int,Na: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B),Na) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_4252_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_4253_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Na) = aa_nat_rat(power_power_rat(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_4254_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_4255_power__eq__iff__eq__base,axiom,
    ! [Na: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(power_power_int(A),Na) = aa_nat_int(power_power_int(B),Na) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_4256_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_4257_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_4258_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_4259_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_4260_self__le__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% self_le_power
tff(fact_4261_self__le__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% self_le_power
tff(fact_4262_self__le__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% self_le_power
tff(fact_4263_self__le__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Na)) ) ) ).

% self_le_power
tff(fact_4264_one__less__power,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)) ) ) ).

% one_less_power
tff(fact_4265_one__less__power,axiom,
    ! [A: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Na)) ) ) ).

% one_less_power
tff(fact_4266_one__less__power,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ).

% one_less_power
tff(fact_4267_one__less__power,axiom,
    ! [A: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Na)) ) ) ).

% one_less_power
tff(fact_4268_card__Diff1__le,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))),finite7802652506058667612T_VEBT(A4)) ).

% card_Diff1_le
tff(fact_4269_card__Diff1__le,axiom,
    ! [A4: set_complex,Xa: complex] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))),finite_card_complex(A4)) ).

% card_Diff1_le
tff(fact_4270_card__Diff1__le,axiom,
    ! [A4: set_literal,Xa: literal] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_literal(minus_7832829386415567259iteral(A4,insert_literal(Xa,bot_bot_set_literal)))),finite_card_literal(A4)) ).

% card_Diff1_le
tff(fact_4271_card__Diff1__le,axiom,
    ! [A4: set_list_nat,Xa: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(Xa,bot_bot_set_list_nat)))),finite_card_list_nat(A4)) ).

% card_Diff1_le
tff(fact_4272_card__Diff1__le,axiom,
    ! [A4: set_set_nat,Xa: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)))),finite_card_set_nat(A4)) ).

% card_Diff1_le
tff(fact_4273_card__Diff1__le,axiom,
    ! [A4: set_int,Xa: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))),finite_card_int(A4)) ).

% card_Diff1_le
tff(fact_4274_card__Diff1__le,axiom,
    ! [A4: set_o,Xa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))),finite_card_o(A4)) ).

% card_Diff1_le
tff(fact_4275_card__Diff1__le,axiom,
    ! [A4: set_nat,Xa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A4)) ).

% card_Diff1_le
tff(fact_4276_card__Diff__subset,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( finite5847741373460823677iteral(B4)
     => ( ord_le7307670543136651348iteral(B4,A4)
       => ( finite_card_literal(minus_7832829386415567259iteral(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_literal(A4)),finite_card_literal(B4)) ) ) ) ).

% card_Diff_subset
tff(fact_4277_card__Diff__subset,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le6045566169113846134st_nat(B4,A4)
       => ( finite_card_list_nat(minus_7954133019191499631st_nat(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4)) ) ) ) ).

% card_Diff_subset
tff(fact_4278_card__Diff__subset,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B4),A4)
       => ( finite_card_set_nat(minus_2163939370556025621et_nat(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4)) ) ) ) ).

% card_Diff_subset
tff(fact_4279_card__Diff__subset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( finite_card_complex(minus_811609699411566653omplex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A4)),finite_card_complex(B4)) ) ) ) ).

% card_Diff_subset
tff(fact_4280_card__Diff__subset,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
       => ( finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4)) ) ) ) ).

% card_Diff_subset
tff(fact_4281_card__Diff__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4)) ) ) ) ).

% card_Diff_subset
tff(fact_4282_card__Diff__subset,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( finite_card_int(minus_minus_set_int(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A4)),finite_card_int(B4)) ) ) ) ).

% card_Diff_subset
tff(fact_4283_diff__card__le__card__Diff,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( finite5847741373460823677iteral(B4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_literal(A4)),finite_card_literal(B4))),finite_card_literal(minus_7832829386415567259iteral(A4,B4))) ) ).

% diff_card_le_card_Diff
tff(fact_4284_diff__card__le__card__Diff,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4))),finite_card_list_nat(minus_7954133019191499631st_nat(A4,B4))) ) ).

% diff_card_le_card_Diff
tff(fact_4285_diff__card__le__card__Diff,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4))),finite_card_set_nat(minus_2163939370556025621et_nat(A4,B4))) ) ).

% diff_card_le_card_Diff
tff(fact_4286_diff__card__le__card__Diff,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A4)),finite_card_int(B4))),finite_card_int(minus_minus_set_int(A4,B4))) ) ).

% diff_card_le_card_Diff
tff(fact_4287_diff__card__le__card__Diff,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A4)),finite_card_complex(B4))),finite_card_complex(minus_811609699411566653omplex(A4,B4))) ) ).

% diff_card_le_card_Diff
tff(fact_4288_diff__card__le__card__Diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4))),finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,B4))) ) ).

% diff_card_le_card_Diff
tff(fact_4289_diff__card__le__card__Diff,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4))),aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,B4))) ) ).

% diff_card_le_card_Diff
tff(fact_4290_card__psubset,axiom,
    ! [B4: set_literal,A4: set_literal] :
      ( finite5847741373460823677iteral(B4)
     => ( ord_le7307670543136651348iteral(A4,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_literal(A4)),finite_card_literal(B4))
         => ord_less_set_literal(A4,B4) ) ) ) ).

% card_psubset
tff(fact_4291_card__psubset,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( finite8100373058378681591st_nat(B4)
     => ( ord_le6045566169113846134st_nat(A4,B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(A4)),finite_card_list_nat(B4))
         => ord_le1190675801316882794st_nat(A4,B4) ) ) ) ).

% card_psubset
tff(fact_4292_card__psubset,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,B4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(A4)),finite_card_set_nat(B4))
         => aa_set_set_nat_o(ord_less_set_set_nat(A4),B4) ) ) ) ).

% card_psubset
tff(fact_4293_card__psubset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B4))
         => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4) ) ) ) ).

% card_psubset
tff(fact_4294_card__psubset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(A4)),finite_card_complex(B4))
         => aa_set_complex_o(ord_less_set_complex(A4),B4) ) ) ) ).

% card_psubset
tff(fact_4295_card__psubset,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(A4)),finite4902975817058060853nteger(B4))
         => aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(A4),B4) ) ) ) ).

% card_psubset
tff(fact_4296_card__psubset,axiom,
    ! [B4: set_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(A4)),finite_card_int(B4))
         => aa_set_int_o(ord_less_set_int(A4),B4) ) ) ) ).

% card_psubset
tff(fact_4297_finite__enumerate__mono,axiom,
    ! [M: nat,Na: nat,S2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_set_nat_o(finite_finite_nat,S2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_set_nat_nat(finite_card_nat,S2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S2),M)),aa_nat_nat(infini8530281810654367211te_nat(S2),Na)) ) ) ) ).

% finite_enumerate_mono
tff(fact_4298_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ) ) ).

% nat_mult_power_less_eq
tff(fact_4299_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S2)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S2))) ) ).

% card_le_Suc_Max
tff(fact_4300_finite__le__enumerate,axiom,
    ! [S2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_set_nat_nat(finite_card_nat,S2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(infini8530281810654367211te_nat(S2),Na)) ) ) ).

% finite_le_enumerate
tff(fact_4301_card__insert__disjoint_H,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((Xa)),A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(aa_set_o_set_o(insert_o((Xa)),A4))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_o(A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4302_card__insert__disjoint_H,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite7802652506058667612T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4))),aa_nat_nat(suc,zero_zero_nat)) = finite7802652506058667612T_VEBT(A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4303_card__insert__disjoint_H,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(Xa),A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(aa_set_real_set_real(insert_real(Xa),A4))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_real(A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4304_card__insert__disjoint_H,axiom,
    ! [A4: set_literal,Xa: literal] :
      ( finite5847741373460823677iteral(A4)
     => ( ~ member_literal(Xa,A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_literal(insert_literal(Xa,A4))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_literal(A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4305_card__insert__disjoint_H,axiom,
    ! [A4: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( ~ member_list_nat(Xa,A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(insert_list_nat(Xa,A4))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_list_nat(A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4306_card__insert__disjoint_H,axiom,
    ! [A4: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( ~ aa_set_set_nat_o(member_set_nat(Xa),A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_set_nat(A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4307_card__insert__disjoint_H,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(insert_nat(Xa),A4))),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(finite_card_nat,A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4308_card__insert__disjoint_H,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(Xa),A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(aa_set_int_set_int(insert_int(Xa),A4))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_int(A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4309_card__insert__disjoint_H,axiom,
    ! [A4: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ~ aa_set_complex_o(member_complex(Xa),A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(insert_complex(Xa,A4))),aa_nat_nat(suc,zero_zero_nat)) = finite_card_complex(A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4310_card__insert__disjoint_H,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite4902975817058060853nteger(insert_Code_integer(Xa,A4))),aa_nat_nat(suc,zero_zero_nat)) = finite4902975817058060853nteger(A4) ) ) ) ).

% card_insert_disjoint'
tff(fact_4311_power__strict__mono,axiom,
    ! [A: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_4312_power__strict__mono,axiom,
    ! [A: rat,B: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Na)),aa_nat_rat(power_power_rat(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_4313_power__strict__mono,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_4314_power__strict__mono,axiom,
    ! [A: int,B: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(B),Na)) ) ) ) ).

% power_strict_mono
tff(fact_4315_card_Oremove,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( finite7802652506058667612T_VEBT(A4) = aa_nat_nat(suc,finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% card.remove
tff(fact_4316_card_Oremove,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => ( finite_card_real(A4) = aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)))) ) ) ) ).

% card.remove
tff(fact_4317_card_Oremove,axiom,
    ! [A4: set_literal,Xa: literal] :
      ( finite5847741373460823677iteral(A4)
     => ( member_literal(Xa,A4)
       => ( finite_card_literal(A4) = aa_nat_nat(suc,finite_card_literal(minus_7832829386415567259iteral(A4,insert_literal(Xa,bot_bot_set_literal)))) ) ) ) ).

% card.remove
tff(fact_4318_card_Oremove,axiom,
    ! [A4: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( member_list_nat(Xa,A4)
       => ( finite_card_list_nat(A4) = aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(Xa,bot_bot_set_list_nat)))) ) ) ) ).

% card.remove
tff(fact_4319_card_Oremove,axiom,
    ! [A4: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A4)
       => ( finite_card_set_nat(A4) = aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)))) ) ) ) ).

% card.remove
tff(fact_4320_card_Oremove,axiom,
    ! [A4: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(Xa),A4)
       => ( finite_card_complex(A4) = aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))) ) ) ) ).

% card.remove
tff(fact_4321_card_Oremove,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( finite4902975817058060853nteger(A4) = aa_nat_nat(suc,finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))) ) ) ) ).

% card.remove
tff(fact_4322_card_Oremove,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( finite_card_int(A4) = aa_nat_nat(suc,finite_card_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))) ) ) ) ).

% card.remove
tff(fact_4323_card_Oremove,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( finite_card_o(A4) = aa_nat_nat(suc,finite_card_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))) ) ) ) ).

% card.remove
tff(fact_4324_card_Oremove,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_set_nat_nat(finite_card_nat,A4) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))) ) ) ) ).

% card.remove
tff(fact_4325_card_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( finite7802652506058667612T_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = aa_nat_nat(suc,finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) ) ) ).

% card.insert_remove
tff(fact_4326_card_Oinsert__remove,axiom,
    ! [A4: set_literal,Xa: literal] :
      ( finite5847741373460823677iteral(A4)
     => ( finite_card_literal(insert_literal(Xa,A4)) = aa_nat_nat(suc,finite_card_literal(minus_7832829386415567259iteral(A4,insert_literal(Xa,bot_bot_set_literal)))) ) ) ).

% card.insert_remove
tff(fact_4327_card_Oinsert__remove,axiom,
    ! [A4: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite_card_list_nat(insert_list_nat(Xa,A4)) = aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(Xa,bot_bot_set_list_nat)))) ) ) ).

% card.insert_remove
tff(fact_4328_card_Oinsert__remove,axiom,
    ! [A4: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( finite_card_set_nat(aa_set1013257548857649334et_nat(insert_set_nat(Xa),A4)) = aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_4329_card_Oinsert__remove,axiom,
    ! [A4: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( finite_card_complex(insert_complex(Xa,A4)) = aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))) ) ) ).

% card.insert_remove
tff(fact_4330_card_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( finite4902975817058060853nteger(insert_Code_integer(Xa,A4)) = aa_nat_nat(suc,finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))) ) ) ).

% card.insert_remove
tff(fact_4331_card_Oinsert__remove,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( finite_card_int(aa_set_int_set_int(insert_int(Xa),A4)) = aa_nat_nat(suc,finite_card_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))) ) ) ).

% card.insert_remove
tff(fact_4332_card_Oinsert__remove,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( finite_card_o(aa_set_o_set_o(insert_o((Xa)),A4)) = aa_nat_nat(suc,finite_card_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))) ) ) ).

% card.insert_remove
tff(fact_4333_card_Oinsert__remove,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(insert_nat(Xa),A4)) = aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))) ) ) ).

% card.insert_remove
tff(fact_4334_card__Suc__Diff1,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( aa_nat_nat(suc,finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) = finite7802652506058667612T_VEBT(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4335_card__Suc__Diff1,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => ( aa_nat_nat(suc,finite_card_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)))) = finite_card_real(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4336_card__Suc__Diff1,axiom,
    ! [A4: set_literal,Xa: literal] :
      ( finite5847741373460823677iteral(A4)
     => ( member_literal(Xa,A4)
       => ( aa_nat_nat(suc,finite_card_literal(minus_7832829386415567259iteral(A4,insert_literal(Xa,bot_bot_set_literal)))) = finite_card_literal(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4337_card__Suc__Diff1,axiom,
    ! [A4: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( member_list_nat(Xa,A4)
       => ( aa_nat_nat(suc,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(Xa,bot_bot_set_list_nat)))) = finite_card_list_nat(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4338_card__Suc__Diff1,axiom,
    ! [A4: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A4)
       => ( aa_nat_nat(suc,finite_card_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)))) = finite_card_set_nat(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4339_card__Suc__Diff1,axiom,
    ! [A4: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(Xa),A4)
       => ( aa_nat_nat(suc,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))) = finite_card_complex(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4340_card__Suc__Diff1,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( aa_nat_nat(suc,finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))) = finite4902975817058060853nteger(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4341_card__Suc__Diff1,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( aa_nat_nat(suc,finite_card_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))) = finite_card_int(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4342_card__Suc__Diff1,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_nat_nat(suc,finite_card_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))) = finite_card_o(A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4343_card__Suc__Diff1,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))) = aa_set_nat_nat(finite_card_nat,A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_4344_card__Diff1__less,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))),finite7802652506058667612T_VEBT(A4)) ) ) ).

% card_Diff1_less
tff(fact_4345_card__Diff1__less,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)))),finite_card_real(A4)) ) ) ).

% card_Diff1_less
tff(fact_4346_card__Diff1__less,axiom,
    ! [A4: set_literal,Xa: literal] :
      ( finite5847741373460823677iteral(A4)
     => ( member_literal(Xa,A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_literal(minus_7832829386415567259iteral(A4,insert_literal(Xa,bot_bot_set_literal)))),finite_card_literal(A4)) ) ) ).

% card_Diff1_less
tff(fact_4347_card__Diff1__less,axiom,
    ! [A4: set_list_nat,Xa: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( member_list_nat(Xa,A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(Xa,bot_bot_set_list_nat)))),finite_card_list_nat(A4)) ) ) ).

% card_Diff1_less
tff(fact_4348_card__Diff1__less,axiom,
    ! [A4: set_set_nat,Xa: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)))),finite_card_set_nat(A4)) ) ) ).

% card_Diff1_less
tff(fact_4349_card__Diff1__less,axiom,
    ! [A4: set_complex,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))),finite_card_complex(A4)) ) ) ).

% card_Diff1_less
tff(fact_4350_card__Diff1__less,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))),finite4902975817058060853nteger(A4)) ) ) ).

% card_Diff1_less
tff(fact_4351_card__Diff1__less,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))),finite_card_int(A4)) ) ) ).

% card_Diff1_less
tff(fact_4352_card__Diff1__less,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((Xa)),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))),finite_card_o(A4)) ) ) ).

% card_Diff1_less
tff(fact_4353_card__Diff1__less,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A4)) ) ) ).

% card_Diff1_less
tff(fact_4354_card__Diff2__less,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Y),bot_bo8194388402131092736T_VEBT)))),finite7802652506058667612T_VEBT(A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4355_card__Diff2__less,axiom,
    ! [A4: set_real,Xa: real,Y: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => ( aa_set_real_o(member_real(Y),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_real(minus_minus_set_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)),aa_set_real_set_real(insert_real(Y),bot_bot_set_real)))),finite_card_real(A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4356_card__Diff2__less,axiom,
    ! [A4: set_literal,Xa: literal,Y: literal] :
      ( finite5847741373460823677iteral(A4)
     => ( member_literal(Xa,A4)
       => ( member_literal(Y,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_literal(minus_7832829386415567259iteral(minus_7832829386415567259iteral(A4,insert_literal(Xa,bot_bot_set_literal)),insert_literal(Y,bot_bot_set_literal)))),finite_card_literal(A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4357_card__Diff2__less,axiom,
    ! [A4: set_list_nat,Xa: list_nat,Y: list_nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( member_list_nat(Xa,A4)
       => ( member_list_nat(Y,A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(Xa,bot_bot_set_list_nat)),insert_list_nat(Y,bot_bot_set_list_nat)))),finite_card_list_nat(A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4358_card__Diff2__less,axiom,
    ! [A4: set_set_nat,Xa: set_nat,Y: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A4)
       => ( aa_set_set_nat_o(member_set_nat(Y),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)),aa_set1013257548857649334et_nat(insert_set_nat(Y),bot_bot_set_set_nat)))),finite_card_set_nat(A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4359_card__Diff2__less,axiom,
    ! [A4: set_complex,Xa: complex,Y: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(Xa),A4)
       => ( aa_set_complex_o(member_complex(Y),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)),insert_complex(Y,bot_bot_set_complex)))),finite_card_complex(A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4360_card__Diff2__less,axiom,
    ! [A4: set_Code_integer,Xa: code_integer,Y: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(Y),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)),insert_Code_integer(Y,bot_bo3990330152332043303nteger)))),finite4902975817058060853nteger(A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4361_card__Diff2__less,axiom,
    ! [A4: set_int,Xa: int,Y: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( aa_set_int_o(member_int(Y),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)),aa_set_int_set_int(insert_int(Y),bot_bot_set_int)))),finite_card_int(A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4362_card__Diff2__less,axiom,
    ! [A4: set_o,Xa: $o,Y: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_set_o_o(member_o((Y)),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)),aa_set_o_set_o(insert_o((Y)),bot_bot_set_o)))),finite_card_o(A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4363_card__Diff2__less,axiom,
    ! [A4: set_nat,Xa: nat,Y: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_set_nat_o(member_nat(Y),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A4)) ) ) ) ).

% card_Diff2_less
tff(fact_4364_card__Diff1__less__iff,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))),finite7802652506058667612T_VEBT(A4))
    <=> ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
        & aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4365_card__Diff1__less__iff,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)))),finite_card_real(A4))
    <=> ( aa_set_real_o(finite_finite_real,A4)
        & aa_set_real_o(member_real(Xa),A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4366_card__Diff1__less__iff,axiom,
    ! [A4: set_literal,Xa: literal] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_literal(minus_7832829386415567259iteral(A4,insert_literal(Xa,bot_bot_set_literal)))),finite_card_literal(A4))
    <=> ( finite5847741373460823677iteral(A4)
        & member_literal(Xa,A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4367_card__Diff1__less__iff,axiom,
    ! [A4: set_list_nat,Xa: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(Xa,bot_bot_set_list_nat)))),finite_card_list_nat(A4))
    <=> ( finite8100373058378681591st_nat(A4)
        & member_list_nat(Xa,A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4368_card__Diff1__less__iff,axiom,
    ! [A4: set_set_nat,Xa: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat)))),finite_card_set_nat(A4))
    <=> ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
        & aa_set_set_nat_o(member_set_nat(Xa),A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4369_card__Diff1__less__iff,axiom,
    ! [A4: set_complex,Xa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))),finite_card_complex(A4))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,A4)
        & aa_set_complex_o(member_complex(Xa),A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4370_card__Diff1__less__iff,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite4902975817058060853nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))),finite4902975817058060853nteger(A4))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
        & aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4371_card__Diff1__less__iff,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))),finite_card_int(A4))
    <=> ( aa_set_int_o(finite_finite_int,A4)
        & aa_set_int_o(member_int(Xa),A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4372_card__Diff1__less__iff,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,finite_card_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))),finite_card_o(A4))
    <=> ( aa_set_o_o(finite_finite_o,A4)
        & aa_set_o_o(member_o((Xa)),A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4373_card__Diff1__less__iff,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))),aa_set_nat_nat(finite_card_nat,A4))
    <=> ( aa_set_nat_o(finite_finite_nat,A4)
        & aa_set_nat_o(member_nat(Xa),A4) ) ) ).

% card_Diff1_less_iff
tff(fact_4374_card__Diff__singleton__if,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT))) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite7802652506058667612T_VEBT(A4)),one_one_nat),finite7802652506058667612T_VEBT(A4)) ) ).

% card_Diff_singleton_if
tff(fact_4375_card__Diff__singleton__if,axiom,
    ! [A4: set_real,Xa: real] :
      ( finite_card_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real))) = $ite(aa_set_real_o(member_real(Xa),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(A4)),one_one_nat),finite_card_real(A4)) ) ).

% card_Diff_singleton_if
tff(fact_4376_card__Diff__singleton__if,axiom,
    ! [A4: set_complex,Xa: complex] :
      ( finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(Xa),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A4)),one_one_nat),finite_card_complex(A4)) ) ).

% card_Diff_singleton_if
tff(fact_4377_card__Diff__singleton__if,axiom,
    ! [A4: set_literal,Xa: literal] :
      ( finite_card_literal(minus_7832829386415567259iteral(A4,insert_literal(Xa,bot_bot_set_literal))) = $ite(member_literal(Xa,A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_literal(A4)),one_one_nat),finite_card_literal(A4)) ) ).

% card_Diff_singleton_if
tff(fact_4378_card__Diff__singleton__if,axiom,
    ! [A4: set_list_nat,Xa: list_nat] :
      ( finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(Xa,bot_bot_set_list_nat))) = $ite(member_list_nat(Xa,A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A4)),one_one_nat),finite_card_list_nat(A4)) ) ).

% card_Diff_singleton_if
tff(fact_4379_card__Diff__singleton__if,axiom,
    ! [A4: set_set_nat,Xa: set_nat] :
      ( finite_card_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat(Xa),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A4)),one_one_nat),finite_card_set_nat(A4)) ) ).

% card_Diff_singleton_if
tff(fact_4380_card__Diff__singleton__if,axiom,
    ! [A4: set_int,Xa: int] :
      ( finite_card_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))) = $ite(aa_set_int_o(member_int(Xa),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A4)),one_one_nat),finite_card_int(A4)) ) ).

% card_Diff_singleton_if
tff(fact_4381_card__Diff__singleton__if,axiom,
    ! [A4: set_o,Xa: $o] :
      ( finite_card_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((Xa)),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(A4)),one_one_nat),finite_card_o(A4)) ) ).

% card_Diff_singleton_if
tff(fact_4382_card__Diff__singleton__if,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(Xa),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A4)),one_one_nat),aa_set_nat_nat(finite_card_nat,A4)) ) ).

% card_Diff_singleton_if
tff(fact_4383_card__Diff__singleton,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( finite7802652506058667612T_VEBT(minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite7802652506058667612T_VEBT(A4)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4384_card__Diff__singleton,axiom,
    ! [Xa: real,A4: set_real] :
      ( aa_set_real_o(member_real(Xa),A4)
     => ( finite_card_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_real(A4)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4385_card__Diff__singleton,axiom,
    ! [Xa: complex,A4: set_complex] :
      ( aa_set_complex_o(member_complex(Xa),A4)
     => ( finite_card_complex(minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_complex(A4)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4386_card__Diff__singleton,axiom,
    ! [Xa: literal,A4: set_literal] :
      ( member_literal(Xa,A4)
     => ( finite_card_literal(minus_7832829386415567259iteral(A4,insert_literal(Xa,bot_bot_set_literal))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_literal(A4)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4387_card__Diff__singleton,axiom,
    ! [Xa: list_nat,A4: set_list_nat] :
      ( member_list_nat(Xa,A4)
     => ( finite_card_list_nat(minus_7954133019191499631st_nat(A4,insert_list_nat(Xa,bot_bot_set_list_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_list_nat(A4)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4388_card__Diff__singleton,axiom,
    ! [Xa: set_nat,A4: set_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),A4)
     => ( finite_card_set_nat(minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(Xa),bot_bot_set_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_set_nat(A4)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4389_card__Diff__singleton,axiom,
    ! [Xa: int,A4: set_int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( finite_card_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_int(A4)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4390_card__Diff__singleton,axiom,
    ! [Xa: $o,A4: set_o] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( finite_card_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,finite_card_o(A4)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4391_card__Diff__singleton,axiom,
    ! [Xa: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_nat(finite_card_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(finite_card_nat,A4)),one_one_nat) ) ) ).

% card_Diff_singleton
tff(fact_4392_the__default_Osimps_I1_J,axiom,
    ! [Uu2: product_prod_nat_nat,Xa: product_prod_nat_nat] : ( the_de3812365490807259288at_nat(Uu2,some_P7363390416028606310at_nat(Xa)) = Xa ) ).

% the_default.simps(1)
tff(fact_4393_the__default_Osimps_I1_J,axiom,
    ! [Uu2: nat,Xa: nat] : ( the_default_nat(Uu2,some_nat(Xa)) = Xa ) ).

% the_default.simps(1)
tff(fact_4394_power__eq__if,axiom,
    ! [P3: rat,M: nat] :
      ( aa_nat_rat(power_power_rat(P3),M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(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_4395_power__eq__if,axiom,
    ! [P3: complex,M: nat] :
      ( aa_nat_complex(power_power_complex(P3),M) = $ite(M = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_4396_power__eq__if,axiom,
    ! [P3: assn,M: nat] :
      ( aa_nat_assn(power_power_assn(P3),M) = $ite(M = zero_zero_nat,one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P3),aa_nat_assn(power_power_assn(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_4397_power__eq__if,axiom,
    ! [P3: uint32,M: nat] :
      ( aa_nat_uint32(power_power_uint32(P3),M) = $ite(M = zero_zero_nat,one_one_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,P3),aa_nat_uint32(power_power_uint32(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_4398_power__eq__if,axiom,
    ! [P3: real,M: nat] :
      ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_4399_power__eq__if,axiom,
    ! [P3: nat,M: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ) ).

% power_eq_if
tff(fact_4400_power__eq__if,axiom,
    ! [P3: int,M: nat] :
      ( aa_nat_int(power_power_int(P3),M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(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_4401_the__default_Osimps_I2_J,axiom,
    ! [Xa: product_prod_nat_nat] : ( the_de3812365490807259288at_nat(Xa,none_P5556105721700978146at_nat) = Xa ) ).

% the_default.simps(2)
tff(fact_4402_the__default_Osimps_I2_J,axiom,
    ! [Xa: nat] : ( the_default_nat(Xa,none_nat) = Xa ) ).

% the_default.simps(2)
tff(fact_4403_power__minus__mult,axiom,
    ! [Na: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Na) ) ) ).

% power_minus_mult
tff(fact_4404_power__minus__mult,axiom,
    ! [Na: nat,A: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_assn(power_power_assn(A),Na) ) ) ).

% power_minus_mult
tff(fact_4405_power__minus__mult,axiom,
    ! [Na: nat,A: uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(power_power_uint32(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_uint32(power_power_uint32(A),Na) ) ) ).

% power_minus_mult
tff(fact_4406_power__minus__mult,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na) ) ) ).

% power_minus_mult
tff(fact_4407_power__minus__mult,axiom,
    ! [Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na) ) ) ).

% power_minus_mult
tff(fact_4408_power__minus__mult,axiom,
    ! [Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),A) = aa_nat_int(power_power_int(A),Na) ) ) ).

% power_minus_mult
tff(fact_4409_finite__enumerate__step,axiom,
    ! [S2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),aa_set_nat_nat(finite_card_nat,S2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(infini8530281810654367211te_nat(S2),Na)),aa_nat_nat(infini8530281810654367211te_nat(S2),aa_nat_nat(suc,Na))) ) ) ).

% finite_enumerate_step
tff(fact_4410_finite__enum__subset,axiom,
    ! [X4: set_nat,Y6: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_set_nat_nat(finite_card_nat,X4))
         => ( aa_nat_nat(infini8530281810654367211te_nat(X4),I2) = aa_nat_nat(infini8530281810654367211te_nat(Y6),I2) ) )
     => ( aa_set_nat_o(finite_finite_nat,X4)
       => ( aa_set_nat_o(finite_finite_nat,Y6)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,X4)),aa_set_nat_nat(finite_card_nat,Y6))
           => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X4),Y6) ) ) ) ) ).

% finite_enum_subset
tff(fact_4411_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4412_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4413_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( 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,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4414_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( 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,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4415_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( 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,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4416_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P2: fun_VEBT_VEBTi_o,Na: nat] :
      ( ! [X: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P2,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s7982070591426661849_VEBTi(Xs))
       => aa_VEBT_VEBTi_o(P2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Na)) ) ) ).

% inthall
tff(fact_4417_inthall,axiom,
    ! [Xs: list_int,P2: fun_int_o,Na: nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),aa_list_int_set_int(set_int2,Xs))
         => aa_int_o(P2,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
       => aa_int_o(P2,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% inthall
tff(fact_4418_inthall,axiom,
    ! [Xs: list_set_nat,P2: fun_set_nat_o,Na: nat] :
      ( ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),set_set_nat2(Xs))
         => aa_set_nat_o(P2,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P2,aa_nat_set_nat(nth_set_nat(Xs),Na)) ) ) ).

% inthall
tff(fact_4419_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o,Na: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
         => aa_VEBT_VEBT_o(P2,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Na)) ) ) ).

% inthall
tff(fact_4420_inthall,axiom,
    ! [Xs: list_real,P2: fun_real_o,Na: nat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs))
         => aa_real_o(P2,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
       => aa_real_o(P2,aa_nat_real(nth_real(Xs),Na)) ) ) ).

% inthall
tff(fact_4421_inthall,axiom,
    ! [Xs: list_o,P2: fun_o_o,Na: nat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),aa_list_o_set_o(set_o2,Xs))
         => aa_o_o(P2,(X)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
       => aa_o_o(P2,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% inthall
tff(fact_4422_inthall,axiom,
    ! [Xs: list_nat,P2: fun_nat_o,Na: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs))
         => aa_nat_o(P2,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
       => aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% inthall
tff(fact_4423_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_s6755466524823107622T_VEBT(Xs))
       => ( size_s6755466524823107622T_VEBT(slice_VEBT_VEBT(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_4424_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_real(Xs))
       => ( size_size_list_real(slice_real(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_4425_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_o(Xs))
       => ( size_size_list_o(slice_o(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_4426_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_list_nat_nat(size_size_list_nat,slice_nat(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_4427_card__Min__le__sum,axiom,
    ! [A4: set_literal,F: fun_literal_nat] :
      ( finite5847741373460823677iteral(A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_literal(A4)),aa_set_nat_nat(lattic8721135487736765967in_nat,image_literal_nat(F,A4)))),groups8652099787943017962al_nat(F,A4)) ) ).

% card_Min_le_sum
tff(fact_4428_card__Min__le__sum,axiom,
    ! [A4: set_list_nat,F: fun_list_nat_nat] :
      ( finite8100373058378681591st_nat(A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_list_nat(A4)),aa_set_nat_nat(lattic8721135487736765967in_nat,image_list_nat_nat(F,A4)))),groups4396056296759096172at_nat(F,A4)) ) ).

% card_Min_le_sum
tff(fact_4429_card__Min__le__sum,axiom,
    ! [A4: set_set_nat,F: fun_set_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_set_nat(A4)),aa_set_nat_nat(lattic8721135487736765967in_nat,image_set_nat_nat(F,A4)))),groups8294997508430121362at_nat(F,A4)) ) ).

% card_Min_le_sum
tff(fact_4430_card__Min__le__sum,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_int(A4)),aa_set_nat_nat(lattic8721135487736765967in_nat,aa_set_int_set_nat(image_int_nat(F),A4)))),groups4541462559716669496nt_nat(F,A4)) ) ).

% card_Min_le_sum
tff(fact_4431_card__Min__le__sum,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_complex(A4)),aa_set_nat_nat(lattic8721135487736765967in_nat,image_complex_nat(F,A4)))),groups5693394587270226106ex_nat(F,A4)) ) ).

% card_Min_le_sum
tff(fact_4432_card__Min__le__sum,axiom,
    ! [A4: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite4902975817058060853nteger(A4)),aa_set_nat_nat(lattic8721135487736765967in_nat,image_951025933927791156er_nat(F,A4)))),groups7237345082560585321er_nat(F,A4)) ) ).

% card_Min_le_sum
tff(fact_4433_card__Min__le__sum,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(lattic8721135487736765967in_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)))),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) ) ).

% card_Min_le_sum
tff(fact_4434_realpow__pos__nth,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),Na) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_4435_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na) = A )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),Na) = A ) )
               => ( Y4 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_4436_length__mul__elem,axiom,
    ! [Xs: list_list_VEBT_VEBT,Na: nat] :
      ( ! [X: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X,set_list_VEBT_VEBT2(Xs))
         => ( size_s6755466524823107622T_VEBT(X) = Na ) )
     => ( size_s6755466524823107622T_VEBT(concat_VEBT_VEBT(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s8217280938318005548T_VEBT(Xs)),Na) ) ) ).

% length_mul_elem
tff(fact_4437_length__mul__elem,axiom,
    ! [Xs: list_list_real,Na: nat] :
      ( ! [X: list_real] :
          ( member_list_real(X,set_list_real2(Xs))
         => ( size_size_list_real(X) = Na ) )
     => ( size_size_list_real(concat_real(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6660260683639930848t_real(Xs)),Na) ) ) ).

% length_mul_elem
tff(fact_4438_length__mul__elem,axiom,
    ! [Xs: list_list_o,Na: nat] :
      ( ! [X: list_o] :
          ( member_list_o(X,set_list_o2(Xs))
         => ( size_size_list_o(X) = Na ) )
     => ( size_size_list_o(concat_o(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s2710708370519433104list_o(Xs)),Na) ) ) ).

% length_mul_elem
tff(fact_4439_length__mul__elem,axiom,
    ! [Xs: list_list_nat,Na: nat] :
      ( ! [X: list_nat] :
          ( member_list_nat(X,set_list_nat2(Xs))
         => ( aa_list_nat_nat(size_size_list_nat,X) = Na ) )
     => ( aa_list_nat_nat(size_size_list_nat,concat_nat(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3023201423986296836st_nat(Xs)),Na) ) ) ).

% length_mul_elem
tff(fact_4440_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat)
     => ( rotate1_VEBT_VEBT(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4441_rotate1__length01,axiom,
    ! [Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),one_one_nat)
     => ( rotate1_real(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4442_rotate1__length01,axiom,
    ! [Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => ( rotate1_o(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4443_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4444_not__real__square__gt__zero,axiom,
    ! [Xa: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa))
    <=> ( Xa = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_4445_slice__complete,axiom,
    ! [Xs: list_VEBT_VEBT] : ( slice_VEBT_VEBT(zero_zero_nat,size_s6755466524823107622T_VEBT(Xs),Xs) = Xs ) ).

% slice_complete
tff(fact_4446_slice__complete,axiom,
    ! [Xs: list_real] : ( slice_real(zero_zero_nat,size_size_list_real(Xs),Xs) = Xs ) ).

% slice_complete
tff(fact_4447_slice__complete,axiom,
    ! [Xs: list_o] : ( slice_o(zero_zero_nat,size_size_list_o(Xs),Xs) = Xs ) ).

% slice_complete
tff(fact_4448_slice__complete,axiom,
    ! [Xs: list_nat] : ( slice_nat(zero_zero_nat,aa_list_nat_nat(size_size_list_nat,Xs),Xs) = Xs ) ).

% slice_complete
tff(fact_4449_real__add__le__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,Xa)) ) ).

% real_add_le_0_iff
tff(fact_4450_real__0__le__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xa)),Y) ) ).

% real_0_le_add_iff
tff(fact_4451_less__eq__real__def,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_real_def
tff(fact_4452_real__arch__pow,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),N)) ) ).

% real_arch_pow
tff(fact_4453_real__arch__pow__inv,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_4454_real__add__less__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xa)) ) ).

% real_add_less_0_iff
tff(fact_4455_real__0__less__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xa)),Y) ) ).

% real_0_less_add_iff
tff(fact_4456_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
           => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4457_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4458_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4459_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
           => ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4460_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4461_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_4462_Skolem__list__nth,axiom,
    ! [K: nat,P2: fun_na154763851189438853EBTi_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: vEBT_VEBTi] : aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P2,I3),X_12) )
    <=> ? [Xs3: list_VEBT_VEBTi] :
          ( ( size_s7982070591426661849_VEBTi(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P2,I3),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_4463_Skolem__list__nth,axiom,
    ! [K: nat,P2: fun_nat_fun_int_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P2,I3),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_int_o(aa_nat_fun_int_o(P2,I3),aa_nat_int(nth_int(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_4464_Skolem__list__nth,axiom,
    ! [K: nat,P2: fun_na6887940067898170868VEBT_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P2,I3),X_12) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P2,I3),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_4465_Skolem__list__nth,axiom,
    ! [K: nat,P2: fun_nat_fun_real_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: real] : aa_real_o(aa_nat_fun_real_o(P2,I3),X_12) )
    <=> ? [Xs3: list_real] :
          ( ( size_size_list_real(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_real_o(aa_nat_fun_real_o(P2,I3),aa_nat_real(nth_real(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_4466_Skolem__list__nth,axiom,
    ! [K: nat,P2: fun_nat_fun_o_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P2,I3),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_o_o(aa_nat_fun_o_o(P2,I3),aa_nat_o(nth_o(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_4467_Skolem__list__nth,axiom,
    ! [K: nat,P2: fun_nat_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,I3),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( aa_list_nat_nat(size_size_list_nat,Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_nat_o(aa_nat_fun_nat_o(P2,I3),aa_nat_nat(nth_nat(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_4468_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( Xs = Ys )
    <=> ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs))
           => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I3) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4469_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4470_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4471_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
           => ( aa_nat_real(nth_real(Xs),I3) = aa_nat_real(nth_real(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4472_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4473_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_4474_obtain__list__from__elements,axiom,
    ! [Na: nat,P2: fun_VE6346650636051888901_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => ? [Li: vEBT_VEBTi] : aa_nat_o(aa_VEB8883426185042993134_nat_o(P2,Li),I2) )
     => ~ ! [L2: list_VEBT_VEBTi] :
            ( ( size_s7982070591426661849_VEBTi(L2) = Na )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
                 => aa_nat_o(aa_VEB8883426185042993134_nat_o(P2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_4475_obtain__list__from__elements,axiom,
    ! [Na: nat,P2: fun_int_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => ? [Li: int] : aa_nat_o(aa_int_fun_nat_o(P2,Li),I2) )
     => ~ ! [L2: list_int] :
            ( ( size_size_list_int(L2) = Na )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
                 => aa_nat_o(aa_int_fun_nat_o(P2,aa_nat_int(nth_int(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_4476_obtain__list__from__elements,axiom,
    ! [Na: nat,P2: fun_VE6480751511521678896_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => ? [Li: vEBT_VEBT] : aa_nat_o(aa_VEB2314620471664397097_nat_o(P2,Li),I2) )
     => ~ ! [L2: list_VEBT_VEBT] :
            ( ( size_s6755466524823107622T_VEBT(L2) = Na )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(P2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_4477_obtain__list__from__elements,axiom,
    ! [Na: nat,P2: fun_real_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => ? [Li: real] : aa_nat_o(aa_real_fun_nat_o(P2,Li),I2) )
     => ~ ! [L2: list_real] :
            ( ( size_size_list_real(L2) = Na )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
                 => aa_nat_o(aa_real_fun_nat_o(P2,aa_nat_real(nth_real(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_4478_obtain__list__from__elements,axiom,
    ! [Na: nat,P2: fun_o_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => ? [Li: $o] : aa_nat_o(aa_o_fun_nat_o(P2,(Li)),I2) )
     => ~ ! [L2: list_o] :
            ( ( size_size_list_o(L2) = Na )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
                 => aa_nat_o(aa_o_fun_nat_o(P2,aa_nat_o(nth_o(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_4479_obtain__list__from__elements,axiom,
    ! [Na: nat,P2: fun_nat_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => ? [Li: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,Li),I2) )
     => ~ ! [L2: list_nat] :
            ( ( aa_list_nat_nat(size_size_list_nat,L2) = Na )
           => ~ ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
                 => aa_nat_o(aa_nat_fun_nat_o(P2,aa_nat_nat(nth_nat(L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_4480_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBTi,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(slice_VEBT_VEBTi(From,To,Xs)),I) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_4481_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_int(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_int(nth_int(slice_int(From,To,Xs)),I) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_4482_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_VEBT_VEBT,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(slice_VEBT_VEBT(From,To,Xs)),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_4483_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_real,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_real(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_real(nth_real(slice_real(From,To,Xs)),I) = aa_nat_real(nth_real(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_4484_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_o,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),size_size_list_o(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_o(nth_o(slice_o(From,To,Xs)),I)
          <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_4485_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,From),To)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,To),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From))
         => ( aa_nat_nat(nth_nat(slice_nat(From,To,Xs)),I) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_4486_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P2: fun_VEBT_VEBTi_o] :
      ( ! [X5: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X5,set_VEBT_VEBTi2(L))
         => aa_VEBT_VEBTi_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(L))
         => aa_VEBT_VEBTi_o(P2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I3)) ) ) ).

% all_set_conv_nth
tff(fact_4487_all__set__conv__nth,axiom,
    ! [L: list_int,P2: fun_int_o] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),aa_list_int_set_int(set_int2,L))
         => aa_int_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(L))
         => aa_int_o(P2,aa_nat_int(nth_int(L),I3)) ) ) ).

% all_set_conv_nth
tff(fact_4488_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L))
         => aa_VEBT_VEBT_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(L))
         => aa_VEBT_VEBT_o(P2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I3)) ) ) ).

% all_set_conv_nth
tff(fact_4489_all__set__conv__nth,axiom,
    ! [L: list_real,P2: fun_real_o] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),aa_lis3769085826154824284t_real(set_real2,L))
         => aa_real_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(L))
         => aa_real_o(P2,aa_nat_real(nth_real(L),I3)) ) ) ).

% all_set_conv_nth
tff(fact_4490_all__set__conv__nth,axiom,
    ! [L: list_o,P2: fun_o_o] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),aa_list_o_set_o(set_o2,L))
         => aa_o_o(P2,(X5)) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(L))
         => aa_o_o(P2,aa_nat_o(nth_o(L),I3)) ) ) ).

% all_set_conv_nth
tff(fact_4491_all__set__conv__nth,axiom,
    ! [L: list_nat,P2: fun_nat_o] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_list_nat_set_nat(set_nat2,L))
         => aa_nat_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,L))
         => aa_nat_o(P2,aa_nat_nat(nth_nat(L),I3)) ) ) ).

% all_set_conv_nth
tff(fact_4492_nth__mem,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Na),set_VEBT_VEBTi2(Xs)) ) ).

% nth_mem
tff(fact_4493_nth__mem,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Na)),aa_list_int_set_int(set_int2,Xs)) ) ).

% nth_mem
tff(fact_4494_nth__mem,axiom,
    ! [Na: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),Na)),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_4495_nth__mem,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Na)),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)) ) ).

% nth_mem
tff(fact_4496_nth__mem,axiom,
    ! [Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),Na)),aa_lis3769085826154824284t_real(set_real2,Xs)) ) ).

% nth_mem
tff(fact_4497_nth__mem,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => aa_set_o_o(member_o(aa_nat_o(nth_o(Xs),Na)),aa_list_o_set_o(set_o2,Xs)) ) ).

% nth_mem
tff(fact_4498_nth__mem,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Na)),aa_list_nat_set_nat(set_nat2,Xs)) ) ).

% nth_mem
tff(fact_4499_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBTi,P2: fun_VEBT_VEBTi_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s7982070591426661849_VEBTi(Xs))
     => ( ! [X: vEBT_VEBTi] :
            ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
           => aa_VEBT_VEBTi_o(P2,X) )
       => aa_VEBT_VEBTi_o(P2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_4500_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_int,P2: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),aa_list_int_set_int(set_int2,Xs))
           => aa_int_o(P2,X) )
       => aa_int_o(P2,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_4501_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
           => aa_VEBT_VEBT_o(P2,X) )
       => aa_VEBT_VEBT_o(P2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_4502_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_real,P2: fun_real_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs))
           => aa_real_o(P2,X) )
       => aa_real_o(P2,aa_nat_real(nth_real(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_4503_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_o,P2: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),aa_list_o_set_o(set_o2,Xs))
           => aa_o_o(P2,(X)) )
       => aa_o_o(P2,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_4504_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),aa_list_nat_set_nat(set_nat2,Xs))
           => aa_nat_o(P2,X) )
       => aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_4505_in__set__conv__nth,axiom,
    ! [Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs))
          & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I3) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_4506_in__set__conv__nth,axiom,
    ! [Xa: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I3) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_4507_in__set__conv__nth,axiom,
    ! [Xa: set_nat,Xs: list_set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I3) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_4508_in__set__conv__nth,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
          & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_4509_in__set__conv__nth,axiom,
    ! [Xa: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I3) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_4510_in__set__conv__nth,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I3)
          <=> (Xa) ) ) ) ).

% in_set_conv_nth
tff(fact_4511_in__set__conv__nth,axiom,
    ! [Xa: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
          & ( aa_nat_nat(nth_nat(Xs),I3) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_4512_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P2: fun_VEBT_VEBTi_o,Xa: vEBT_VEBTi] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)) )
     => ( member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(Xs))
       => aa_VEBT_VEBTi_o(P2,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_4513_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P2: fun_int_o,Xa: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
         => aa_int_o(P2,aa_nat_int(nth_int(Xs),I2)) )
     => ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
       => aa_int_o(P2,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_4514_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P2: fun_set_nat_o,Xa: set_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P2,aa_nat_set_nat(nth_set_nat(Xs),I2)) )
     => ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
       => aa_set_nat_o(P2,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_4515_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o,Xa: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
       => aa_VEBT_VEBT_o(P2,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_4516_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P2: fun_real_o,Xa: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P2,aa_nat_real(nth_real(Xs),I2)) )
     => ( aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,Xs))
       => aa_real_o(P2,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_4517_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P2: fun_o_o,Xa: $o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
         => aa_o_o(P2,aa_nat_o(nth_o(Xs),I2)) )
     => ( aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,Xs))
       => aa_o_o(P2,(Xa)) ) ) ).

% all_nth_imp_all_set
tff(fact_4518_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P2: fun_nat_o,Xa: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),I2)) )
     => ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs))
       => aa_nat_o(P2,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_4519_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P2: fun_VEBT_VEBTi_o] :
      ( ! [X5: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X5,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_4520_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P2: fun_int_o] :
      ( ! [X5: int] :
          ( aa_set_int_o(member_int(X5),aa_list_int_set_int(set_int2,Xs))
         => aa_int_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
         => aa_int_o(P2,aa_nat_int(nth_int(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_4521_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( ! [X5: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
         => aa_VEBT_VEBT_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_4522_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P2: fun_real_o] :
      ( ! [X5: real] :
          ( aa_set_real_o(member_real(X5),aa_lis3769085826154824284t_real(set_real2,Xs))
         => aa_real_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
         => aa_real_o(P2,aa_nat_real(nth_real(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_4523_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P2: fun_o_o] :
      ( ! [X5: $o] :
          ( aa_set_o_o(member_o((X5)),aa_list_o_set_o(set_o2,Xs))
         => aa_o_o(P2,(X5)) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
         => aa_o_o(P2,aa_nat_o(nth_o(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_4524_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P2: fun_nat_o] :
      ( ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_list_nat_set_nat(set_nat2,Xs))
         => aa_nat_o(P2,X5) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(P2,aa_nat_nat(nth_nat(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_4525_realpow__pos__nth2,axiom,
    ! [A: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,Na)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_4526_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : ( aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero_nat ) ).

% VEBT.size(4)
tff(fact_4527_set__image__eq__pointwiseI,axiom,
    ! [L: list_int,L3: list_int,F: fun_int_nat] :
      ( ( size_size_list_int(L) = size_size_list_int(L3) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(L))
           => ( aa_int_nat(F,aa_nat_int(nth_int(L),I2)) = aa_int_nat(F,aa_nat_int(nth_int(L3),I2)) ) )
       => ( aa_set_int_set_nat(image_int_nat(F),aa_list_int_set_int(set_int2,L)) = aa_set_int_set_nat(image_int_nat(F),aa_list_int_set_int(set_int2,L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_4528_set__image__eq__pointwiseI,axiom,
    ! [L: list_int,L3: list_int,F: fun_int_int] :
      ( ( size_size_list_int(L) = size_size_list_int(L3) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(L))
           => ( aa_int_int(F,aa_nat_int(nth_int(L),I2)) = aa_int_int(F,aa_nat_int(nth_int(L3),I2)) ) )
       => ( aa_set_int_set_int(image_int_int(F),aa_list_int_set_int(set_int2,L)) = aa_set_int_set_int(image_int_int(F),aa_list_int_set_int(set_int2,L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_4529_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L3: list_nat,F: fun_nat_set_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,L) = aa_list_nat_nat(size_size_list_nat,L3) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,L))
           => ( aa_nat_set_nat(F,aa_nat_nat(nth_nat(L),I2)) = aa_nat_set_nat(F,aa_nat_nat(nth_nat(L3),I2)) ) )
       => ( aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_list_nat_set_nat(set_nat2,L)) = aa_set7246812951530217472et_nat(image_nat_set_nat(F),aa_list_nat_set_nat(set_nat2,L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_4530_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L3: list_nat,F: fun_nat_nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,L) = aa_list_nat_nat(size_size_list_nat,L3) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,L))
           => ( aa_nat_nat(F,aa_nat_nat(nth_nat(L),I2)) = aa_nat_nat(F,aa_nat_nat(nth_nat(L3),I2)) ) )
       => ( aa_set_nat_set_nat(image_nat_nat(F),aa_list_nat_set_nat(set_nat2,L)) = aa_set_nat_set_nat(image_nat_nat(F),aa_list_nat_set_nat(set_nat2,L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_4531_set__image__eq__pointwiseI,axiom,
    ! [L: list_nat,L3: list_nat,F: fun_nat_int] :
      ( ( aa_list_nat_nat(size_size_list_nat,L) = aa_list_nat_nat(size_size_list_nat,L3) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,L))
           => ( aa_nat_int(F,aa_nat_nat(nth_nat(L),I2)) = aa_nat_int(F,aa_nat_nat(nth_nat(L3),I2)) ) )
       => ( aa_set_nat_set_int(image_nat_int(F),aa_list_nat_set_nat(set_nat2,L)) = aa_set_nat_set_int(image_nat_int(F),aa_list_nat_set_nat(set_nat2,L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_4532_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBTi_nat,Xa: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( aa_set_nat_o(member_nat(aa_VEBT_VEBTi_nat(F,Xa)),image_VEBT_VEBTi_nat(F,set_VEBT_VEBTi2(L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(L))
          & ( aa_VEBT_VEBTi_nat(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I3)) = aa_VEBT_VEBTi_nat(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4533_in__set__image__conv__nth,axiom,
    ! [F: fun_int_nat,Xa: int,L: list_int] :
      ( aa_set_nat_o(member_nat(aa_int_nat(F,Xa)),aa_set_int_set_nat(image_int_nat(F),aa_list_int_set_int(set_int2,L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(L))
          & ( aa_int_nat(F,aa_nat_int(nth_int(L),I3)) = aa_int_nat(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4534_in__set__image__conv__nth,axiom,
    ! [F: fun_VE9120655849212850072T_VEBT,Xa: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB4941942288398148609T_VEBT(F,Xa)),image_7547481670047419768T_VEBT(F,set_VEBT_VEBTi2(L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(L))
          & ( aa_VEB4941942288398148609T_VEBT(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I3)) = aa_VEB4941942288398148609T_VEBT(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4535_in__set__image__conv__nth,axiom,
    ! [F: fun_int_VEBT_VEBT,Xa: int,L: list_int] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_int_VEBT_VEBT(F,Xa)),image_int_VEBT_VEBT(F,aa_list_int_set_int(set_int2,L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(L))
          & ( aa_int_VEBT_VEBT(F,aa_nat_int(nth_int(L),I3)) = aa_int_VEBT_VEBT(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4536_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBTi_real,Xa: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( aa_set_real_o(member_real(aa_VEBT_VEBTi_real(F,Xa)),image_6202559892754154600i_real(F,set_VEBT_VEBTi2(L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(L))
          & ( aa_VEBT_VEBTi_real(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I3)) = aa_VEBT_VEBTi_real(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4537_in__set__image__conv__nth,axiom,
    ! [F: fun_int_real,Xa: int,L: list_int] :
      ( aa_set_real_o(member_real(aa_int_real(F,Xa)),image_int_real(F,aa_list_int_set_int(set_int2,L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(L))
          & ( aa_int_real(F,aa_nat_int(nth_int(L),I3)) = aa_int_real(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4538_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBTi_int,Xa: vEBT_VEBTi,L: list_VEBT_VEBTi] :
      ( aa_set_int_o(member_int(aa_VEBT_VEBTi_int(F,Xa)),image_VEBT_VEBTi_int(F,set_VEBT_VEBTi2(L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(L))
          & ( aa_VEBT_VEBTi_int(F,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I3)) = aa_VEBT_VEBTi_int(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4539_in__set__image__conv__nth,axiom,
    ! [F: fun_int_int,Xa: int,L: list_int] :
      ( aa_set_int_o(member_int(aa_int_int(F,Xa)),aa_set_int_set_int(image_int_int(F),aa_list_int_set_int(set_int2,L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(L))
          & ( aa_int_int(F,aa_nat_int(nth_int(L),I3)) = aa_int_int(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4540_in__set__image__conv__nth,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xa: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(aa_VEBT_VEBT_nat(F,Xa)),image_VEBT_VEBT_nat(F,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(L))
          & ( aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I3)) = aa_VEBT_VEBT_nat(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4541_in__set__image__conv__nth,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xa: vEBT_VEBT,L: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(F,Xa)),image_3375948659692109573T_VEBT(F,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(L))
          & ( aa_VEB5832953974749077948T_VEBT(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I3)) = aa_VEB5832953974749077948T_VEBT(F,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4542_sum__count__set,axiom,
    ! [Xs: list_complex,X4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),X4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,X4)
       => ( groups5693394587270226106ex_nat(count_list_complex(Xs),X4) = size_s3451745648224563538omplex(Xs) ) ) ) ).

% sum_count_set
tff(fact_4543_sum__count__set,axiom,
    ! [Xs: list_Code_integer,X4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Xs)),X4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X4)
       => ( groups7237345082560585321er_nat(count_3970941599679287265nteger(Xs),X4) = size_s3445333598471063425nteger(Xs) ) ) ) ).

% sum_count_set
tff(fact_4544_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),X4)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,X4)
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X4) = size_s6755466524823107622T_VEBT(Xs) ) ) ) ).

% sum_count_set
tff(fact_4545_sum__count__set,axiom,
    ! [Xs: list_real,X4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),X4)
     => ( aa_set_real_o(finite_finite_real,X4)
       => ( groups1935376822645274424al_nat(count_list_real(Xs),X4) = size_size_list_real(Xs) ) ) ) ).

% sum_count_set
tff(fact_4546_sum__count__set,axiom,
    ! [Xs: list_o,X4: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Xs)),X4)
     => ( aa_set_o_o(finite_finite_o,X4)
       => ( groups8507830703676809646_o_nat(count_list_o(Xs),X4) = size_size_list_o(Xs) ) ) ) ).

% sum_count_set
tff(fact_4547_sum__count__set,axiom,
    ! [Xs: list_int,X4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Xs)),X4)
     => ( aa_set_int_o(finite_finite_int,X4)
       => ( groups4541462559716669496nt_nat(count_list_int(Xs),X4) = size_size_list_int(Xs) ) ) ) ).

% sum_count_set
tff(fact_4548_sum__count__set,axiom,
    ! [Xs: list_nat,X4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Xs)),X4)
     => ( aa_set_nat_o(finite_finite_nat,X4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(count_list_nat(Xs)),X4) = aa_list_nat_nat(size_size_list_nat,Xs) ) ) ) ).

% sum_count_set
tff(fact_4549_sum__le__card__Max,axiom,
    ! [A4: set_literal,F: fun_literal_nat] :
      ( finite5847741373460823677iteral(A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8652099787943017962al_nat(F,A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_literal(A4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_literal_nat(F,A4)))) ) ).

% sum_le_card_Max
tff(fact_4550_sum__le__card__Max,axiom,
    ! [A4: set_list_nat,F: fun_list_nat_nat] :
      ( finite8100373058378681591st_nat(A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4396056296759096172at_nat(F,A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_list_nat(A4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_list_nat_nat(F,A4)))) ) ).

% sum_le_card_Max
tff(fact_4551_sum__le__card__Max,axiom,
    ! [A4: set_set_nat,F: fun_set_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups8294997508430121362at_nat(F,A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_set_nat(A4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_set_nat_nat(F,A4)))) ) ).

% sum_le_card_Max
tff(fact_4552_sum__le__card__Max,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_int(A4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_int_set_nat(image_int_nat(F),A4)))) ) ).

% sum_le_card_Max
tff(fact_4553_sum__le__card__Max,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite_card_complex(A4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_complex_nat(F,A4)))) ) ).

% sum_le_card_Max
tff(fact_4554_sum__le__card__Max,axiom,
    ! [A4: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups7237345082560585321er_nat(F,A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,finite4902975817058060853nteger(A4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_951025933927791156er_nat(F,A4)))) ) ).

% sum_le_card_Max
tff(fact_4555_sum__le__card__Max,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)))) ) ).

% sum_le_card_Max
tff(fact_4556_sum_Oinsert,axiom,
    ! [A4: set_o,Xa: $o,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((Xa)),A4)
       => ( groups8691415230153176458o_real(G,aa_set_o_set_o(insert_o((Xa)),A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_o_real(G,(Xa))),groups8691415230153176458o_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4557_sum_Oinsert,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( groups2240296850493347238T_real(G,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xa)),groups2240296850493347238T_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4558_sum_Oinsert,axiom,
    ! [A4: set_real,Xa: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(Xa),A4)
       => ( groups8097168146408367636l_real(G,aa_set_real_set_real(insert_real(Xa),A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xa)),groups8097168146408367636l_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4559_sum_Oinsert,axiom,
    ! [A4: set_int,Xa: int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(Xa),A4)
       => ( groups8778361861064173332t_real(G,aa_set_int_set_int(insert_int(Xa),A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xa)),groups8778361861064173332t_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4560_sum_Oinsert,axiom,
    ! [A4: set_complex,Xa: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ~ aa_set_complex_o(member_complex(Xa),A4)
       => ( groups5808333547571424918x_real(G,insert_complex(Xa,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xa)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4561_sum_Oinsert,axiom,
    ! [A4: set_Code_integer,Xa: code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( groups1270011288395367621r_real(G,insert_Code_integer(Xa,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,Xa)),groups1270011288395367621r_real(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4562_sum_Oinsert,axiom,
    ! [A4: set_o,Xa: $o,G: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((Xa)),A4)
       => ( groups7872700643590313910_o_rat(G,aa_set_o_set_o(insert_o((Xa)),A4)) = aa_rat_rat(plus_plus_rat(aa_o_rat(G,(Xa))),groups7872700643590313910_o_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4563_sum_Oinsert,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( groups136491112297645522BT_rat(G,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xa)),groups136491112297645522BT_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4564_sum_Oinsert,axiom,
    ! [A4: set_real,Xa: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(Xa),A4)
       => ( groups1300246762558778688al_rat(G,aa_set_real_set_real(insert_real(Xa),A4)) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xa)),groups1300246762558778688al_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4565_sum_Oinsert,axiom,
    ! [A4: set_nat,Xa: nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(Xa),A4)
       => ( groups2906978787729119204at_rat(G,aa_set_nat_set_nat(insert_nat(Xa),A4)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xa)),groups2906978787729119204at_rat(G,A4)) ) ) ) ).

% sum.insert
tff(fact_4566_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_uint32] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups833757482993574392uint32(G,A4) = zero_zero_uint32 ) ) ).

% sum.infinite
tff(fact_4567_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_uint32] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups5712668689793887828uint32(G,A4) = zero_zero_uint32 ) ) ).

% sum.infinite
tff(fact_4568_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_uint32] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups8736914816313324502uint32(G,A4) = zero_zero_uint32 ) ) ).

% sum.infinite
tff(fact_4569_sum_Oinfinite,axiom,
    ! [A4: set_Code_integer,G: fun_Co1554742298434871494uint32] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups8847630953604152069uint32(G,A4) = zero_zero_uint32 ) ) ).

% sum.infinite
tff(fact_4570_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(G,A4) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_4571_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,A4) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_4572_sum_Oinfinite,axiom,
    ! [A4: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups1270011288395367621r_real(G,A4) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_4573_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( groups2906978787729119204at_rat(G,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_4574_sum_Oinfinite,axiom,
    ! [A4: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(G,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_4575_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(G,A4) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_4576_sum__eq__0__iff,axiom,
    ! [F2: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,F2)
     => ( ( groups4541462559716669496nt_nat(F,F2) = zero_zero_nat )
      <=> ! [X5: int] :
            ( aa_set_int_o(member_int(X5),F2)
           => ( aa_int_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4577_sum__eq__0__iff,axiom,
    ! [F2: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F2)
     => ( ( groups5693394587270226106ex_nat(F,F2) = zero_zero_nat )
      <=> ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),F2)
           => ( aa_complex_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4578_sum__eq__0__iff,axiom,
    ! [F2: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F2)
     => ( ( groups7237345082560585321er_nat(F,F2) = zero_zero_nat )
      <=> ! [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),F2)
           => ( aa_Code_integer_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4579_sum__eq__0__iff,axiom,
    ! [F2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,F2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F2) = zero_zero_nat )
      <=> ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),F2)
           => ( aa_nat_nat(F,X5) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_4580_sum_Oempty,axiom,
    ! [G: fun_nat_uint32] : ( groups833757482993574392uint32(G,bot_bot_set_nat) = zero_zero_uint32 ) ).

% sum.empty
tff(fact_4581_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : ( groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ) ).

% sum.empty
tff(fact_4582_sum_Oempty,axiom,
    ! [G: fun_nat_int] : ( aa_set_nat_int(groups3539618377306564664at_int(G),bot_bot_set_nat) = zero_zero_int ) ).

% sum.empty
tff(fact_4583_sum_Oempty,axiom,
    ! [G: fun_int_uint32] : ( groups5712668689793887828uint32(G,bot_bot_set_int) = zero_zero_uint32 ) ).

% sum.empty
tff(fact_4584_sum_Oempty,axiom,
    ! [G: fun_int_real] : ( groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ) ).

% sum.empty
tff(fact_4585_sum_Oempty,axiom,
    ! [G: fun_int_rat] : ( groups3906332499630173760nt_rat(G,bot_bot_set_int) = zero_zero_rat ) ).

% sum.empty
tff(fact_4586_sum_Oempty,axiom,
    ! [G: fun_int_nat] : ( groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ) ).

% sum.empty
tff(fact_4587_sum_Oempty,axiom,
    ! [G: fun_o_uint32] : ( groups7241207224191747786uint32(G,bot_bot_set_o) = zero_zero_uint32 ) ).

% sum.empty
tff(fact_4588_sum_Oempty,axiom,
    ! [G: fun_o_real] : ( groups8691415230153176458o_real(G,bot_bot_set_o) = zero_zero_real ) ).

% sum.empty
tff(fact_4589_sum_Oempty,axiom,
    ! [G: fun_o_rat] : ( groups7872700643590313910_o_rat(G,bot_bot_set_o) = zero_zero_rat ) ).

% sum.empty
tff(fact_4590_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A4)
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X)) )
       => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),groups2240296850493347238T_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4591_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: fun_real_real] :
      ( aa_set_real_o(member_real(I),A4)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),minus_minus_set_real(A4,aa_set_real_set_real(insert_real(I),bot_bot_set_real)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
       => ( aa_set_real_o(finite_finite_real,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4592_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(member_complex(I),A4)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(A4,insert_complex(I,bot_bot_set_complex)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4593_member__le__sum,axiom,
    ! [I: code_integer,A4: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),A4)
     => ( ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(A4,insert_Code_integer(I,bot_bo3990330152332043303nteger)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,X)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,I)),groups1270011288395367621r_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4594_member__le__sum,axiom,
    ! [I: int,A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(member_int(I),A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),minus_minus_set_int(A4,aa_set_int_set_int(insert_int(I),bot_bot_set_int)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X)) )
       => ( aa_set_int_o(finite_finite_int,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4595_member__le__sum,axiom,
    ! [I: $o,A4: set_o,F: fun_o_real] :
      ( aa_set_o_o(member_o((I)),A4)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),minus_minus_set_o(A4,aa_set_o_set_o(insert_o((I)),bot_bot_set_o)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_o_real(F,(X))) )
       => ( aa_set_o_o(finite_finite_o,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F,(I))),groups8691415230153176458o_real(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4596_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A4)
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(I),bot_bo8194388402131092736T_VEBT)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X)) )
       => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),groups136491112297645522BT_rat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4597_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: fun_real_rat] :
      ( aa_set_real_o(member_real(I),A4)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),minus_minus_set_real(A4,aa_set_real_set_real(insert_real(I),bot_bot_set_real)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X)) )
       => ( aa_set_real_o(finite_finite_real,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4598_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(member_complex(I),A4)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(A4,insert_complex(I,bot_bot_set_complex)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4599_member__le__sum,axiom,
    ! [I: code_integer,A4: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),A4)
     => ( ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(A4,insert_Code_integer(I,bot_bo3990330152332043303nteger)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,X)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,I)),groups6602215022474089585er_rat(F,A4)) ) ) ) ).

% member_le_sum
tff(fact_4600_sum__strict__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,B))
           => ( ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2240296850493347238T_real(F,A4)),groups2240296850493347238T_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4601_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,B))
           => ( ! [X: real] :
                  ( aa_set_real_o(member_real(X),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4602_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,B))
           => ( ! [X: complex] :
                  ( aa_set_complex_o(member_complex(X),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4603_sum__strict__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,B: code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B4,A4))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Code_integer_real(F,B))
           => ( ! [X: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,X)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1270011288395367621r_real(F,A4)),groups1270011288395367621r_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4604_sum__strict__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B))
           => ( ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups136491112297645522BT_rat(F,A4)),groups136491112297645522BT_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4605_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( aa_set_real_o(member_real(B),minus_minus_set_real(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,B))
           => ( ! [X: real] :
                  ( aa_set_real_o(member_real(X),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4606_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B))
           => ( ! [X: complex] :
                  ( aa_set_complex_o(member_complex(X),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4607_sum__strict__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,B: code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(F,B))
           => ( ! [X: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,X)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(F,A4)),groups6602215022474089585er_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4608_sum__strict__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,B: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B4,A4))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B))
           => ( ! [X: nat] :
                  ( aa_set_nat_o(member_nat(X),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4609_sum__strict__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B: vEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(B4,A4))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B))
           => ( ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups771621172384141258BT_nat(F,A4)),groups771621172384141258BT_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4610_sum__diff1,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_uint32,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups8325533452322294502uint32(F,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4),minus_minus_uint32(groups8325533452322294502uint32(F,A4),aa_VEBT_VEBT_uint32(F,A)),groups8325533452322294502uint32(F,A4)) ) ) ).

% sum_diff1
tff(fact_4611_sum__diff1,axiom,
    ! [A4: set_real,F: fun_real_uint32,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups5944083974425963860uint32(F,minus_minus_set_real(A4,aa_set_real_set_real(insert_real(A),bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A4),minus_minus_uint32(groups5944083974425963860uint32(F,A4),aa_real_uint32(F,A)),groups5944083974425963860uint32(F,A4)) ) ) ).

% sum_diff1
tff(fact_4612_sum__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_uint32,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups8736914816313324502uint32(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A4),minus_minus_uint32(groups8736914816313324502uint32(F,A4),aa_complex_uint32(F,A)),groups8736914816313324502uint32(F,A4)) ) ) ).

% sum_diff1
tff(fact_4613_sum__diff1,axiom,
    ! [A4: set_Code_integer,F: fun_Co1554742298434871494uint32,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups8847630953604152069uint32(F,minus_2355218937544613996nteger(A4,insert_Code_integer(A,bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),A4),minus_minus_uint32(groups8847630953604152069uint32(F,A4),aa_Cod8401701657498852871uint32(F,A)),groups8847630953604152069uint32(F,A4)) ) ) ).

% sum_diff1
tff(fact_4614_sum__diff1,axiom,
    ! [A4: set_int,F: fun_int_uint32,A: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups5712668689793887828uint32(F,minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A4),minus_minus_uint32(groups5712668689793887828uint32(F,A4),aa_int_uint32(F,A)),groups5712668689793887828uint32(F,A4)) ) ) ).

% sum_diff1
tff(fact_4615_sum__diff1,axiom,
    ! [A4: set_o,F: fun_o_uint32,A: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7241207224191747786uint32(F,minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A4),minus_minus_uint32(groups7241207224191747786uint32(F,A4),aa_o_uint32(F,(A))),groups7241207224191747786uint32(F,A4)) ) ) ).

% sum_diff1
tff(fact_4616_sum__diff1,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,A: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups2240296850493347238T_real(F,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4),aa_real_real(minus_minus_real(groups2240296850493347238T_real(F,A4)),aa_VEBT_VEBT_real(F,A)),groups2240296850493347238T_real(F,A4)) ) ) ).

% sum_diff1
tff(fact_4617_sum__diff1,axiom,
    ! [A4: set_real,F: fun_real_real,A: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups8097168146408367636l_real(F,minus_minus_set_real(A4,aa_set_real_set_real(insert_real(A),bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A4),aa_real_real(minus_minus_real(groups8097168146408367636l_real(F,A4)),aa_real_real(F,A)),groups8097168146408367636l_real(F,A4)) ) ) ).

% sum_diff1
tff(fact_4618_sum__diff1,axiom,
    ! [A4: set_complex,F: fun_complex_real,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,insert_complex(A,bot_bot_set_complex))) = $ite(aa_set_complex_o(member_complex(A),A4),aa_real_real(minus_minus_real(groups5808333547571424918x_real(F,A4)),aa_complex_real(F,A)),groups5808333547571424918x_real(F,A4)) ) ) ).

% sum_diff1
tff(fact_4619_sum__diff1,axiom,
    ! [A4: set_Code_integer,F: fun_Co3960896333048504454r_real,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups1270011288395367621r_real(F,minus_2355218937544613996nteger(A4,insert_Code_integer(A,bot_bo3990330152332043303nteger))) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),A4),aa_real_real(minus_minus_real(groups1270011288395367621r_real(F,A4)),aa_Code_integer_real(F,A)),groups1270011288395367621r_real(F,A4)) ) ) ).

% sum_diff1
tff(fact_4620_sum_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups2240296850493347238T_real(G,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xa)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_4621_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_real,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,insert_complex(Xa,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xa)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_4622_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,G: fun_Co3960896333048504454r_real,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups1270011288395367621r_real(G,insert_Code_integer(Xa,A4)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,Xa)),groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_4623_sum_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups136491112297645522BT_rat(G,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xa)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_4624_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_rat,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(G,insert_complex(Xa,A4)) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xa)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_4625_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,G: fun_Code_integer_rat,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(G,insert_Code_integer(Xa,A4)) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(G,Xa)),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_4626_sum_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups771621172384141258BT_nat(G,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xa)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_4627_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: fun_complex_nat,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5693394587270226106ex_nat(G,insert_complex(Xa,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,Xa)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))) ) ) ).

% sum.insert_remove
tff(fact_4628_sum_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,G: fun_Code_integer_nat,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups7237345082560585321er_nat(G,insert_Code_integer(Xa,A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(G,Xa)),groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.insert_remove
tff(fact_4629_sum_Oinsert__remove,axiom,
    ! [A4: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups769130701875090982BT_int(G,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,Xa)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.insert_remove
tff(fact_4630_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X3: real] : aa_set_real_o(member_real(X3),S2)
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa_set_real_o(member_real(X),S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z4) )
       => ? [Y3: real] :
            ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),S2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y3) )
            & ! [Z4: real] :
                ( ! [X: real] :
                    ( aa_set_real_o(member_real(X),S2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Z4) ) ) ) ) ).

% complete_real
tff(fact_4631_real__minus__mult__self__le,axiom,
    ! [U: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)) ).

% real_minus_mult_self_le
tff(fact_4632_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A4)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X)),A4)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X)) = aa_nat_nat(G,aa_nat_nat(suc,X)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) ) ) ) ).

% sum_cong_Suc
tff(fact_4633_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A4)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X)),A4)
           => ( aa_nat_real(F,aa_nat_nat(suc,X)) = aa_nat_real(G,aa_nat_nat(suc,X)) ) )
       => ( groups6591440286371151544t_real(F,A4) = groups6591440286371151544t_real(G,A4) ) ) ) ).

% sum_cong_Suc
tff(fact_4634_sum_Oneutral,axiom,
    ! [A4: set_nat,G: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => ( aa_nat_nat(G,X) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A4) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_4635_sum_Oneutral,axiom,
    ! [A4: set_complex,G: fun_complex_complex] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),A4)
         => ( aa_complex_complex(G,X) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A4) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_4636_sum_Oneutral,axiom,
    ! [A4: set_nat,G: fun_nat_real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => ( aa_nat_real(G,X) = zero_zero_real ) )
     => ( groups6591440286371151544t_real(G,A4) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_4637_sum_Oneutral,axiom,
    ! [A4: set_int,G: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => ( aa_int_int(G,X) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A4) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_4638_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_uint32,A4: set_nat] :
      ( ( groups833757482993574392uint32(G,A4) != zero_zero_uint32 )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ( aa_nat_uint32(G,A3) = zero_zero_uint32 ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4639_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_uint32,A4: set_VEBT_VEBT] :
      ( ( groups8325533452322294502uint32(G,A4) != zero_zero_uint32 )
     => ~ ! [A3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A4)
           => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4640_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_uint32,A4: set_real] :
      ( ( groups5944083974425963860uint32(G,A4) != zero_zero_uint32 )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A4)
           => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4641_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_uint32,A4: set_int] :
      ( ( groups5712668689793887828uint32(G,A4) != zero_zero_uint32 )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ( aa_int_uint32(G,A3) = zero_zero_uint32 ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4642_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_real,A4: set_VEBT_VEBT] :
      ( ( groups2240296850493347238T_real(G,A4) != zero_zero_real )
     => ~ ! [A3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A4)
           => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4643_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A4: set_real] :
      ( ( groups8097168146408367636l_real(G,A4) != zero_zero_real )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A4)
           => ( aa_real_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4644_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A4: set_int] :
      ( ( groups8778361861064173332t_real(G,A4) != zero_zero_real )
     => ~ ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => ( aa_int_real(G,A3) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4645_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A4: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => ( aa_nat_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4646_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_rat,A4: set_VEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A4)
           => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4647_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A4: set_real] :
      ( ( groups1300246762558778688al_rat(G,A4) != zero_zero_rat )
     => ~ ! [A3: real] :
            ( aa_set_real_o(member_real(A3),A4)
           => ( aa_real_rat(G,A3) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4648_sum__nonpos,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,X)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2240296850493347238T_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_4649_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_4650_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A4)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_4651_sum__nonpos,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_4652_sum__nonpos,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,X)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups136491112297645522BT_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_4653_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_4654_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A4)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_4655_sum__nonpos,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,X)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups771621172384141258BT_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_4656_sum__nonpos,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_4657_sum__nonpos,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A4)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_4658_sum__nonneg,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups2240296850493347238T_real(F,A4)) ) ).

% sum_nonneg
tff(fact_4659_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A4)) ) ).

% sum_nonneg
tff(fact_4660_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A4)) ) ).

% sum_nonneg
tff(fact_4661_sum__nonneg,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_4662_sum__nonneg,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups136491112297645522BT_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_4663_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_4664_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A4)) ) ).

% sum_nonneg
tff(fact_4665_sum__nonneg,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups771621172384141258BT_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_4666_sum__nonneg,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_4667_sum__nonneg,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A4)) ) ).

% sum_nonneg
tff(fact_4668_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_rat,I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(F,I5) = groups136491112297645522BT_rat(G,I5) )
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I2)),aa_VEBT_VEBT_rat(G,I2)) )
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
         => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
           => ( aa_VEBT_VEBT_rat(F,I) = aa_VEBT_VEBT_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4669_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I5: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I5) = groups1300246762558778688al_rat(G,I5) )
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) )
       => ( aa_set_real_o(member_real(I),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4670_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I5) = groups2906978787729119204at_rat(G,I5) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4671_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I5: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G,I5) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4672_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I5: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I5) = groups5058264527183730370ex_rat(G,I5) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4673_sum__mono__inv,axiom,
    ! [F: fun_Code_integer_rat,I5: set_Code_integer,G: fun_Code_integer_rat,I: code_integer] :
      ( ( groups6602215022474089585er_rat(F,I5) = groups6602215022474089585er_rat(G,I5) )
     => ( ! [I2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,I2)),aa_Code_integer_rat(G,I2)) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(I),I5)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
           => ( aa_Code_integer_rat(F,I) = aa_Code_integer_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4674_sum__mono__inv,axiom,
    ! [F: fun_VEBT_VEBT_nat,I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,I: vEBT_VEBT] :
      ( ( groups771621172384141258BT_nat(F,I5) = groups771621172384141258BT_nat(G,I5) )
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I2)),aa_VEBT_VEBT_nat(G,I2)) )
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
         => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
           => ( aa_VEBT_VEBT_nat(F,I) = aa_VEBT_VEBT_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4675_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I5: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G,I5) )
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
       => ( aa_set_real_o(member_real(I),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4676_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4677_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I5: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G,I5) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4678_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X)) )
       => ( ( groups2240296850493347238T_real(F,A4) = zero_zero_real )
        <=> ! [X5: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),A4)
             => ( aa_VEBT_VEBT_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4679_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
       => ( ( groups8097168146408367636l_real(F,A4) = zero_zero_real )
        <=> ! [X5: real] :
              ( aa_set_real_o(member_real(X5),A4)
             => ( aa_real_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4680_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X)) )
       => ( ( groups8778361861064173332t_real(F,A4) = zero_zero_real )
        <=> ! [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
             => ( aa_int_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4681_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X)) )
       => ( ( groups5808333547571424918x_real(F,A4) = zero_zero_real )
        <=> ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),A4)
             => ( aa_complex_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4682_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,X)) )
       => ( ( groups1270011288395367621r_real(F,A4) = zero_zero_real )
        <=> ! [X5: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
             => ( aa_Code_integer_real(F,X5) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4683_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X)) )
       => ( ( groups136491112297645522BT_rat(F,A4) = zero_zero_rat )
        <=> ! [X5: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),A4)
             => ( aa_VEBT_VEBT_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4684_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X)) )
       => ( ( groups1300246762558778688al_rat(F,A4) = zero_zero_rat )
        <=> ! [X5: real] :
              ( aa_set_real_o(member_real(X5),A4)
             => ( aa_real_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4685_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X)) )
       => ( ( groups2906978787729119204at_rat(F,A4) = zero_zero_rat )
        <=> ! [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),A4)
             => ( aa_nat_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4686_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X)) )
       => ( ( groups3906332499630173760nt_rat(F,A4) = zero_zero_rat )
        <=> ! [X5: int] :
              ( aa_set_int_o(member_int(X5),A4)
             => ( aa_int_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4687_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X)) )
       => ( ( groups5058264527183730370ex_rat(F,A4) = zero_zero_rat )
        <=> ! [X5: complex] :
              ( aa_set_complex_o(member_complex(X5),A4)
             => ( aa_complex_rat(F,X5) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4688_sum__le__included,axiom,
    ! [Sa: set_int,T: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,Sa)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X)) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),Sa)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),T)
                    & ( aa_int_int(I,Xa2) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),aa_int_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,Sa)),groups8778361861064173332t_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4689_sum__le__included,axiom,
    ! [Sa: set_int,T: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,Sa)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X)) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),Sa)
               => ? [Xa2: complex] :
                    ( aa_set_complex_o(member_complex(Xa2),T)
                    & ( aa_complex_int(I,Xa2) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),aa_complex_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,Sa)),groups5808333547571424918x_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4690_sum__le__included,axiom,
    ! [Sa: set_int,T: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Code_integer_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,Sa)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X)) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),Sa)
               => ? [Xa2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),T)
                    & ( aa_Code_integer_int(I,Xa2) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),aa_Code_integer_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,Sa)),groups1270011288395367621r_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4691_sum__le__included,axiom,
    ! [Sa: set_complex,T: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sa)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X)) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),Sa)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),T)
                    & ( aa_int_complex(I,Xa2) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X)),aa_int_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,Sa)),groups8778361861064173332t_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4692_sum__le__included,axiom,
    ! [Sa: set_complex,T: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sa)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X)) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),Sa)
               => ? [Xa2: complex] :
                    ( aa_set_complex_o(member_complex(Xa2),T)
                    & ( aa_complex_complex(I,Xa2) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X)),aa_complex_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,Sa)),groups5808333547571424918x_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4693_sum__le__included,axiom,
    ! [Sa: set_complex,T: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Co1900716531650368264omplex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sa)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X)) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),Sa)
               => ? [Xa2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),T)
                    & ( aa_Cod5621350732091595593omplex(I,Xa2) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X)),aa_Code_integer_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,Sa)),groups1270011288395367621r_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4694_sum__le__included,axiom,
    ! [Sa: set_Code_integer,T: set_int,G: fun_int_real,I: fun_int_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Sa)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X)) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),Sa)
               => ? [Xa2: int] :
                    ( aa_set_int_o(member_int(Xa2),T)
                    & ( aa_int_Code_integer(I,Xa2) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,X)),aa_int_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(F,Sa)),groups8778361861064173332t_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4695_sum__le__included,axiom,
    ! [Sa: set_Code_integer,T: set_complex,G: fun_complex_real,I: fun_co8153928105989009288nteger,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Sa)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X)) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),Sa)
               => ? [Xa2: complex] :
                    ( aa_set_complex_o(member_complex(Xa2),T)
                    & ( aa_com4217951221296830281nteger(I,Xa2) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,X)),aa_complex_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(F,Sa)),groups5808333547571424918x_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4696_sum__le__included,axiom,
    ! [Sa: set_Code_integer,T: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Co8499238673658658103nteger,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Sa)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),T)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X)) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),Sa)
               => ? [Xa2: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),T)
                    & ( aa_Cod1783974652533396728nteger(I,Xa2) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,X)),aa_Code_integer_real(G,Xa2)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(F,Sa)),groups1270011288395367621r_real(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4697_sum__le__included,axiom,
    ! [Sa: set_nat,T: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,Sa)
     => ( aa_set_nat_o(finite_finite_nat,T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),T)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X)) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),Sa)
               => ? [Xa2: nat] :
                    ( aa_set_nat_o(member_nat(Xa2),T)
                    & ( aa_nat_nat(I,Xa2) = X )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(G,Xa2)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,Sa)),groups2906978787729119204at_rat(G,T)) ) ) ) ) ).

% sum_le_included
tff(fact_4698_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),aa_int_real(G,X)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4699_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X)),aa_complex_real(G,X)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4700_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,X)),aa_Code_integer_real(G,X)) )
       => ( ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Code_integer_real(F,X3)),aa_Code_integer_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1270011288395367621r_real(F,A4)),groups1270011288395367621r_real(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4701_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(G,X)) )
       => ( ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4702_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),aa_int_rat(G,X)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A4)),groups3906332499630173760nt_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4703_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X)),aa_complex_rat(G,X)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4704_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F,X)),aa_Code_integer_rat(G,X)) )
       => ( ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(F,X3)),aa_Code_integer_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(F,A4)),groups6602215022474089585er_rat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4705_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),aa_int_nat(G,X)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4706_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X)),aa_complex_nat(G,X)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4707_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,X)),aa_Code_integer_nat(G,X)) )
       => ( ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,X3)),aa_Code_integer_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups7237345082560585321er_nat(F,A4)),groups7237345082560585321er_nat(G,A4)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4708_sum_Orelated,axiom,
    ! [R2: fun_ui3478071520421546738nt32_o,S2: set_nat,H2: fun_nat_uint32,G: fun_nat_uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,zero_zero_uint32),zero_zero_uint32)
     => ( ! [X15: uint32,Y15: uint32,X23: uint32,Y23: uint32] :
            ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,X15),X23)
              & aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,Y15),Y23) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,aa_uint32_uint32(plus_plus_uint32(X15),Y15)),aa_uint32_uint32(plus_plus_uint32(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,aa_nat_uint32(H2,X)),aa_nat_uint32(G,X)) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,groups833757482993574392uint32(H2,S2)),groups833757482993574392uint32(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4709_sum_Orelated,axiom,
    ! [R2: fun_ui3478071520421546738nt32_o,S2: set_int,H2: fun_int_uint32,G: fun_int_uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,zero_zero_uint32),zero_zero_uint32)
     => ( ! [X15: uint32,Y15: uint32,X23: uint32,Y23: uint32] :
            ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,X15),X23)
              & aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,Y15),Y23) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,aa_uint32_uint32(plus_plus_uint32(X15),Y15)),aa_uint32_uint32(plus_plus_uint32(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,aa_int_uint32(H2,X)),aa_int_uint32(G,X)) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,groups5712668689793887828uint32(H2,S2)),groups5712668689793887828uint32(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4710_sum_Orelated,axiom,
    ! [R2: fun_ui3478071520421546738nt32_o,S2: set_complex,H2: fun_complex_uint32,G: fun_complex_uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,zero_zero_uint32),zero_zero_uint32)
     => ( ! [X15: uint32,Y15: uint32,X23: uint32,Y23: uint32] :
            ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,X15),X23)
              & aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,Y15),Y23) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,aa_uint32_uint32(plus_plus_uint32(X15),Y15)),aa_uint32_uint32(plus_plus_uint32(X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,aa_complex_uint32(H2,X)),aa_complex_uint32(G,X)) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,groups8736914816313324502uint32(H2,S2)),groups8736914816313324502uint32(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4711_sum_Orelated,axiom,
    ! [R2: fun_ui3478071520421546738nt32_o,S2: set_Code_integer,H2: fun_Co1554742298434871494uint32,G: fun_Co1554742298434871494uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,zero_zero_uint32),zero_zero_uint32)
     => ( ! [X15: uint32,Y15: uint32,X23: uint32,Y23: uint32] :
            ( ( aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,X15),X23)
              & aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,Y15),Y23) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,aa_uint32_uint32(plus_plus_uint32(X15),Y15)),aa_uint32_uint32(plus_plus_uint32(X23),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S2)
               => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,aa_Cod8401701657498852871uint32(H2,X)),aa_Cod8401701657498852871uint32(G,X)) )
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(R2,groups8847630953604152069uint32(H2,S2)),groups8847630953604152069uint32(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4712_sum_Orelated,axiom,
    ! [R2: fun_real_fun_real_o,S2: set_int,H2: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R2,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R2,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R2,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_real_o(aa_real_fun_real_o(R2,aa_int_real(H2,X)),aa_int_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R2,groups8778361861064173332t_real(H2,S2)),groups8778361861064173332t_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4713_sum_Orelated,axiom,
    ! [R2: fun_real_fun_real_o,S2: set_complex,H2: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R2,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R2,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R2,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_real_o(aa_real_fun_real_o(R2,aa_complex_real(H2,X)),aa_complex_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R2,groups5808333547571424918x_real(H2,S2)),groups5808333547571424918x_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4714_sum_Orelated,axiom,
    ! [R2: fun_real_fun_real_o,S2: set_Code_integer,H2: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_real_o(aa_real_fun_real_o(R2,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R2,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R2,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S2)
               => aa_real_o(aa_real_fun_real_o(R2,aa_Code_integer_real(H2,X)),aa_Code_integer_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R2,groups1270011288395367621r_real(H2,S2)),groups1270011288395367621r_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4715_sum_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_o,S2: set_nat,H2: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R2,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R2,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R2,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R2,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R2,aa_nat_rat(H2,X)),aa_nat_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R2,groups2906978787729119204at_rat(H2,S2)),groups2906978787729119204at_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4716_sum_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_o,S2: set_int,H2: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R2,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R2,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R2,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R2,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R2,aa_int_rat(H2,X)),aa_int_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R2,groups3906332499630173760nt_rat(H2,S2)),groups3906332499630173760nt_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4717_sum_Orelated,axiom,
    ! [R2: fun_rat_fun_rat_o,S2: set_complex,H2: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R2,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R2,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R2,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R2,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R2,aa_complex_rat(H2,X)),aa_complex_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R2,groups5058264527183730370ex_rat(H2,S2)),groups5058264527183730370ex_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_4718_sum__strict__mono,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( ( A4 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_VEBT_VEBT_real(F,X)),aa_VEBT_VEBT_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2240296850493347238T_real(F,A4)),groups2240296850493347238T_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4719_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4720_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X)),aa_complex_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4721_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,F: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Code_integer_real(F,X)),aa_Code_integer_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1270011288395367621r_real(F,A4)),groups1270011288395367621r_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4722_sum__strict__mono,axiom,
    ! [A4: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4723_sum__strict__mono,axiom,
    ! [A4: set_o,F: fun_o_real,G: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X: $o] :
              ( aa_set_o_o(member_o((X)),A4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_o_real(F,(X))),aa_o_real(G,(X))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8691415230153176458o_real(F,A4)),groups8691415230153176458o_real(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4724_sum__strict__mono,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( ( A4 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_VEBT_VEBT_rat(F,X)),aa_VEBT_VEBT_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups136491112297645522BT_rat(F,A4)),groups136491112297645522BT_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4725_sum__strict__mono,axiom,
    ! [A4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4726_sum__strict__mono,axiom,
    ! [A4: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X)),aa_complex_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4727_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,F: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),A4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(F,X)),aa_Code_integer_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(F,A4)),groups6602215022474089585er_rat(G,A4)) ) ) ) ).

% sum_strict_mono
tff(fact_4728_sum_Oinsert__if,axiom,
    ! [A4: set_o,G: fun_o_real,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups8691415230153176458o_real(G,aa_set_o_set_o(insert_o((Xa)),A4)) = $ite(aa_set_o_o(member_o((Xa)),A4),groups8691415230153176458o_real(G,A4),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_o_real(G,(Xa))),groups8691415230153176458o_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4729_sum_Oinsert__if,axiom,
    ! [A4: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups2240296850493347238T_real(G,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4),groups2240296850493347238T_real(G,A4),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xa)),groups2240296850493347238T_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4730_sum_Oinsert__if,axiom,
    ! [A4: set_real,G: fun_real_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups8097168146408367636l_real(G,aa_set_real_set_real(insert_real(Xa),A4)) = $ite(aa_set_real_o(member_real(Xa),A4),groups8097168146408367636l_real(G,A4),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xa)),groups8097168146408367636l_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4731_sum_Oinsert__if,axiom,
    ! [A4: set_int,G: fun_int_real,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(G,aa_set_int_set_int(insert_int(Xa),A4)) = $ite(aa_set_int_o(member_int(Xa),A4),groups8778361861064173332t_real(G,A4),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,Xa)),groups8778361861064173332t_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4732_sum_Oinsert__if,axiom,
    ! [A4: set_complex,G: fun_complex_real,Xa: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,insert_complex(Xa,A4)) = $ite(aa_set_complex_o(member_complex(Xa),A4),groups5808333547571424918x_real(G,A4),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xa)),groups5808333547571424918x_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4733_sum_Oinsert__if,axiom,
    ! [A4: set_Code_integer,G: fun_Co3960896333048504454r_real,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups1270011288395367621r_real(G,insert_Code_integer(Xa,A4)) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4),groups1270011288395367621r_real(G,A4),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,Xa)),groups1270011288395367621r_real(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4734_sum_Oinsert__if,axiom,
    ! [A4: set_o,G: fun_o_rat,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( groups7872700643590313910_o_rat(G,aa_set_o_set_o(insert_o((Xa)),A4)) = $ite(aa_set_o_o(member_o((Xa)),A4),groups7872700643590313910_o_rat(G,A4),aa_rat_rat(plus_plus_rat(aa_o_rat(G,(Xa))),groups7872700643590313910_o_rat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4735_sum_Oinsert__if,axiom,
    ! [A4: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,Xa: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups136491112297645522BT_rat(G,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),A4)) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4),groups136491112297645522BT_rat(G,A4),aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xa)),groups136491112297645522BT_rat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4736_sum_Oinsert__if,axiom,
    ! [A4: set_real,G: fun_real_rat,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups1300246762558778688al_rat(G,aa_set_real_set_real(insert_real(Xa),A4)) = $ite(aa_set_real_o(member_real(Xa),A4),groups1300246762558778688al_rat(G,A4),aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xa)),groups1300246762558778688al_rat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4737_sum_Oinsert__if,axiom,
    ! [A4: set_nat,G: fun_nat_rat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups2906978787729119204at_rat(G,aa_set_nat_set_nat(insert_nat(Xa),A4)) = $ite(aa_set_nat_o(member_nat(Xa),A4),groups2906978787729119204at_rat(G,A4),aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Xa)),groups2906978787729119204at_rat(G,A4))) ) ) ).

% sum.insert_if
tff(fact_4738_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_VEBT_VEBT,T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T2: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,H2: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T5)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T2,T5)) )
           => ( ! [B3: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T2,T5))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T2,T5))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S5)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S5)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),T5)
                       => ( aa_VEBT_VEBT_uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_VEBT_VEBT_uint32(H2,aa_VEB5832953974749077948T_VEBT(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups8325533452322294502uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4739_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_VEBT_VEBT,T5: set_real,S2: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T2: set_real,G: fun_VEBT_VEBT_uint32,H2: fun_real_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
     => ( aa_set_real_o(finite_finite_real,T5)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
               => aa_set_real_o(member_real(aa_VEBT_VEBT_real(J,A3)),minus_minus_set_real(T2,T5)) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(T2,T5))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( aa_set_real_o(member_real(B3),minus_minus_set_real(T2,T5))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S5)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S5)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: real] :
                        ( aa_set_real_o(member_real(B3),T5)
                       => ( aa_real_uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_real_uint32(H2,aa_VEBT_VEBT_real(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups5944083974425963860uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4740_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_VEBT_VEBT,S2: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T2: set_VEBT_VEBT,G: fun_real_uint32,H2: fun_VEBT_VEBT_uint32] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(J,A3)),minus_5127226145743854075T_VEBT(T2,T5)) )
           => ( ! [B3: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T2,T5))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B3)) = B3 ) )
             => ( ! [B3: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(T2,T5))
                   => aa_set_real_o(member_real(aa_VEBT_VEBT_real(I,B3)),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S5)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),T5)
                       => ( aa_VEBT_VEBT_uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_VEBT_VEBT_uint32(H2,aa_real_VEBT_VEBT(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups8325533452322294502uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4741_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_uint32,H2: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set_real_o(finite_finite_real,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
             => ( aa_real_real(I,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
               => aa_set_real_o(member_real(aa_real_real(J,A3)),minus_minus_set_real(T2,T5)) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(T2,T5))
                 => ( aa_real_real(J,aa_real_real(I,B3)) = B3 ) )
             => ( ! [B3: real] :
                    ( aa_set_real_o(member_real(B3),minus_minus_set_real(T2,T5))
                   => aa_set_real_o(member_real(aa_real_real(I,B3)),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S5)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: real] :
                        ( aa_set_real_o(member_real(B3),T5)
                       => ( aa_real_uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_real_uint32(H2,aa_real_real(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups5944083974425963860uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4742_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_VEBT_VEBT,T5: set_int,S2: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T2: set_int,G: fun_VEBT_VEBT_uint32,H2: fun_int_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
     => ( aa_set_int_o(finite_finite_int,T5)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
               => aa_set_int_o(member_int(aa_VEBT_VEBT_int(J,A3)),minus_minus_set_int(T2,T5)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T2,T5))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T2,T5))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_int_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S5)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S5)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T5)
                       => ( aa_int_uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_int_uint32(H2,aa_VEBT_VEBT_int(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups5712668689793887828uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4743_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_uint32,H2: fun_int_uint32] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set_int_o(finite_finite_int,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
             => ( aa_int_real(I,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
               => aa_set_int_o(member_int(aa_real_int(J,A3)),minus_minus_set_int(T2,T5)) )
           => ( ! [B3: int] :
                  ( aa_set_int_o(member_int(B3),minus_minus_set_int(T2,T5))
                 => ( aa_real_int(J,aa_int_real(I,B3)) = B3 ) )
             => ( ! [B3: int] :
                    ( aa_set_int_o(member_int(B3),minus_minus_set_int(T2,T5))
                   => aa_set_real_o(member_real(aa_int_real(I,B3)),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S5)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: int] :
                        ( aa_set_int_o(member_int(B3),T5)
                       => ( aa_int_uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_int_uint32(H2,aa_real_int(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups5712668689793887828uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4744_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_VEBT_VEBT,T5: set_complex,S2: set_VEBT_VEBT,I: fun_co4340422964966740447T_VEBT,J: fun_VE5569070535936029429omplex,T2: set_complex,G: fun_VEBT_VEBT_uint32,H2: fun_complex_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T5)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
             => ( aa_complex_VEBT_VEBT(I,aa_VEBT_VEBT_complex(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
               => aa_set_complex_o(member_complex(aa_VEBT_VEBT_complex(J,A3)),minus_811609699411566653omplex(T2,T5)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T2,T5))
                 => ( aa_VEBT_VEBT_complex(J,aa_complex_VEBT_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T2,T5))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_complex_VEBT_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S5)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S5)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T5)
                       => ( aa_complex_uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_complex_uint32(H2,aa_VEBT_VEBT_complex(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups8736914816313324502uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4745_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_uint32,H2: fun_complex_uint32] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
             => ( aa_complex_real(I,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A3)),minus_811609699411566653omplex(T2,T5)) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T2,T5))
                 => ( aa_real_complex(J,aa_complex_real(I,B3)) = B3 ) )
             => ( ! [B3: complex] :
                    ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(T2,T5))
                   => aa_set_real_o(member_real(aa_complex_real(I,B3)),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S5)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: complex] :
                        ( aa_set_complex_o(member_complex(B3),T5)
                       => ( aa_complex_uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_complex_uint32(H2,aa_real_complex(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups8736914816313324502uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4746_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_VEBT_VEBT,T5: set_Code_integer,S2: set_VEBT_VEBT,I: fun_Co3766629010929971184T_VEBT,J: fun_VE9219535225822561060nteger,T2: set_Code_integer,G: fun_VEBT_VEBT_uint32,H2: fun_Co1554742298434871494uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
       => ( ! [A3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
             => ( aa_Cod6507154402108480601T_VEBT(I,aa_VEB5935388676523804373nteger(J,A3)) = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(S2,S5))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_VEB5935388676523804373nteger(J,A3)),minus_2355218937544613996nteger(T2,T5)) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(T2,T5))
                 => ( aa_VEB5935388676523804373nteger(J,aa_Cod6507154402108480601T_VEBT(I,B3)) = B3 ) )
             => ( ! [B3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(T2,T5))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_Cod6507154402108480601T_VEBT(I,B3)),minus_5127226145743854075T_VEBT(S2,S5)) )
               => ( ! [A3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S5)
                     => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B3),T5)
                       => ( aa_Cod8401701657498852871uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),S2)
                         => ( aa_Cod8401701657498852871uint32(H2,aa_VEB5935388676523804373nteger(J,A3)) = aa_VEBT_VEBT_uint32(G,A3) ) )
                     => ( groups8325533452322294502uint32(G,S2) = groups8847630953604152069uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4747_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_Code_integer,S2: set_real,I: fun_Co3960896333048504454r_real,J: fun_re6978948838062957830nteger,T2: set_Code_integer,G: fun_real_uint32,H2: fun_Co1554742298434871494uint32] :
      ( aa_set_real_o(finite_finite_real,S5)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
       => ( ! [A3: real] :
              ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
             => ( aa_Code_integer_real(I,aa_real_Code_integer(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(S2,S5))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_real_Code_integer(J,A3)),minus_2355218937544613996nteger(T2,T5)) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(T2,T5))
                 => ( aa_real_Code_integer(J,aa_Code_integer_real(I,B3)) = B3 ) )
             => ( ! [B3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(T2,T5))
                   => aa_set_real_o(member_real(aa_Code_integer_real(I,B3)),minus_minus_set_real(S2,S5)) )
               => ( ! [A3: real] :
                      ( aa_set_real_o(member_real(A3),S5)
                     => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
                 => ( ! [B3: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B3),T5)
                       => ( aa_Cod8401701657498852871uint32(H2,B3) = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( aa_set_real_o(member_real(A3),S2)
                         => ( aa_Cod8401701657498852871uint32(H2,aa_real_Code_integer(J,A3)) = aa_real_uint32(G,A3) ) )
                     => ( groups5944083974425963860uint32(G,S2) = groups8847630953604152069uint32(H2,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_4748_sum__SucD,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Na: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,Na) )
     => ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X)) ) ) ).

% sum_SucD
tff(fact_4749_sum__eq__Suc0__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( groups4541462559716669496nt_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
            & ( aa_int_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A4)
               => ( ( X5 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4750_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( groups5693394587270226106ex_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
            & ( aa_complex_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A4)
               => ( ( X5 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4751_sum__eq__Suc0__iff,axiom,
    ! [A4: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( groups7237345082560585321er_nat(F,A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
            & ( aa_Code_integer_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A4)
               => ( ( X5 != Xa3 )
                 => ( aa_Code_integer_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4752_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
            & ( aa_nat_nat(F,X5) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A4)
               => ( ( X5 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4753_sum__eq__1__iff,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( groups4541462559716669496nt_nat(F,A4) = one_one_nat )
      <=> ? [X5: int] :
            ( aa_set_int_o(member_int(X5),A4)
            & ( aa_int_nat(F,X5) = one_one_nat )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A4)
               => ( ( X5 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4754_sum__eq__1__iff,axiom,
    ! [A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( groups5693394587270226106ex_nat(F,A4) = one_one_nat )
      <=> ? [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),A4)
            & ( aa_complex_nat(F,X5) = one_one_nat )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A4)
               => ( ( X5 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4755_sum__eq__1__iff,axiom,
    ! [A4: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( groups7237345082560585321er_nat(F,A4) = one_one_nat )
      <=> ? [X5: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X5),A4)
            & ( aa_Code_integer_nat(F,X5) = one_one_nat )
            & ! [Xa3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa3),A4)
               => ( ( X5 != Xa3 )
                 => ( aa_Code_integer_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4756_sum__eq__1__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A4) = one_one_nat )
      <=> ? [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),A4)
            & ( aa_nat_nat(F,X5) = one_one_nat )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A4)
               => ( ( X5 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4757_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I))
         => ( ! [I2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4758_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4759_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4760_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4761_sum__pos2,axiom,
    ! [I5: set_Code_integer,I: code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Code_integer_real(F,I))
         => ( ! [I2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups1270011288395367621r_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4762_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I))
         => ( ! [I2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4763_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4764_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4765_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4766_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_4767_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups2240296850493347238T_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4768_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4769_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4770_sum__pos,axiom,
    ! [I5: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Code_integer_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups1270011288395367621r_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4771_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4772_sum__pos,axiom,
    ! [I5: set_o,F: fun_o_real] :
      ( aa_set_o_o(finite_finite_o,I5)
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( aa_set_o_o(member_o((I2)),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_o_real(F,(I2))) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8691415230153176458o_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4773_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4774_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4775_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4776_sum__pos,axiom,
    ! [I5: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups6602215022474089585er_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_4777_sum_Osame__carrier,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,H2: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),C3)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C3)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C3,A4))
               => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C3,B4))
                 => ( aa_VEBT_VEBT_uint32(H2,B3) = zero_zero_uint32 ) )
             => ( ( groups8325533452322294502uint32(G,A4) = groups8325533452322294502uint32(H2,B4) )
              <=> ( groups8325533452322294502uint32(G,C3) = groups8325533452322294502uint32(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4778_sum_Osame__carrier,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: fun_real_uint32,H2: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,C3)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C3)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C3)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C3,A4))
               => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C3,B4))
                 => ( aa_real_uint32(H2,B3) = zero_zero_uint32 ) )
             => ( ( groups5944083974425963860uint32(G,A4) = groups5944083974425963860uint32(H2,B4) )
              <=> ( groups5944083974425963860uint32(G,C3) = groups5944083974425963860uint32(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4779_sum_Osame__carrier,axiom,
    ! [C3: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_uint32,H2: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C3)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C3)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C3,A4))
               => ( aa_complex_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C3,B4))
                 => ( aa_complex_uint32(H2,B3) = zero_zero_uint32 ) )
             => ( ( groups8736914816313324502uint32(G,A4) = groups8736914816313324502uint32(H2,B4) )
              <=> ( groups8736914816313324502uint32(G,C3) = groups8736914816313324502uint32(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4780_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,G: fun_Co1554742298434871494uint32,H2: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Cod8401701657498852871uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Cod8401701657498852871uint32(H2,B3) = zero_zero_uint32 ) )
             => ( ( groups8847630953604152069uint32(G,A4) = groups8847630953604152069uint32(H2,B4) )
              <=> ( groups8847630953604152069uint32(G,C3) = groups8847630953604152069uint32(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4781_sum_Osame__carrier,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H2: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),C3)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C3)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C3,A4))
               => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C3,B4))
                 => ( aa_VEBT_VEBT_real(H2,B3) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,A4) = groups2240296850493347238T_real(H2,B4) )
              <=> ( groups2240296850493347238T_real(G,C3) = groups2240296850493347238T_real(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4782_sum_Osame__carrier,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: fun_real_real,H2: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C3)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C3)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C3)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C3,A4))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C3,B4))
                 => ( aa_real_real(H2,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H2,B4) )
              <=> ( groups8097168146408367636l_real(G,C3) = groups8097168146408367636l_real(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4783_sum_Osame__carrier,axiom,
    ! [C3: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H2: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C3)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C3)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C3,A4))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C3,B4))
                 => ( aa_complex_real(H2,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H2,B4) )
              <=> ( groups5808333547571424918x_real(G,C3) = groups5808333547571424918x_real(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4784_sum_Osame__carrier,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,G: fun_Co3960896333048504454r_real,H2: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_real(H2,B3) = zero_zero_real ) )
             => ( ( groups1270011288395367621r_real(G,A4) = groups1270011288395367621r_real(H2,B4) )
              <=> ( groups1270011288395367621r_real(G,C3) = groups1270011288395367621r_real(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4785_sum_Osame__carrier,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H2: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),C3)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C3)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C3,A4))
               => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C3,B4))
                 => ( aa_VEBT_VEBT_rat(H2,B3) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,A4) = groups136491112297645522BT_rat(H2,B4) )
              <=> ( groups136491112297645522BT_rat(G,C3) = groups136491112297645522BT_rat(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4786_sum_Osame__carrier,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H2: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C3)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C3)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C3)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C3,A4))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C3,B4))
                 => ( aa_real_rat(H2,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H2,B4) )
              <=> ( groups1300246762558778688al_rat(G,C3) = groups1300246762558778688al_rat(H2,C3) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4787_sum_Osame__carrierI,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,H2: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),C3)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C3)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C3,A4))
               => ( aa_VEBT_VEBT_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C3,B4))
                 => ( aa_VEBT_VEBT_uint32(H2,B3) = zero_zero_uint32 ) )
             => ( ( groups8325533452322294502uint32(G,C3) = groups8325533452322294502uint32(H2,C3) )
               => ( groups8325533452322294502uint32(G,A4) = groups8325533452322294502uint32(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4788_sum_Osame__carrierI,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: fun_real_uint32,H2: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,C3)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C3)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C3)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C3,A4))
               => ( aa_real_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C3,B4))
                 => ( aa_real_uint32(H2,B3) = zero_zero_uint32 ) )
             => ( ( groups5944083974425963860uint32(G,C3) = groups5944083974425963860uint32(H2,C3) )
               => ( groups5944083974425963860uint32(G,A4) = groups5944083974425963860uint32(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4789_sum_Osame__carrierI,axiom,
    ! [C3: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_uint32,H2: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C3)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C3)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C3,A4))
               => ( aa_complex_uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C3,B4))
                 => ( aa_complex_uint32(H2,B3) = zero_zero_uint32 ) )
             => ( ( groups8736914816313324502uint32(G,C3) = groups8736914816313324502uint32(H2,C3) )
               => ( groups8736914816313324502uint32(G,A4) = groups8736914816313324502uint32(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4790_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,G: fun_Co1554742298434871494uint32,H2: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Cod8401701657498852871uint32(G,A3) = zero_zero_uint32 ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Cod8401701657498852871uint32(H2,B3) = zero_zero_uint32 ) )
             => ( ( groups8847630953604152069uint32(G,C3) = groups8847630953604152069uint32(H2,C3) )
               => ( groups8847630953604152069uint32(G,A4) = groups8847630953604152069uint32(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4791_sum_Osame__carrierI,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H2: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),C3)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C3)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C3,A4))
               => ( aa_VEBT_VEBT_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C3,B4))
                 => ( aa_VEBT_VEBT_real(H2,B3) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,C3) = groups2240296850493347238T_real(H2,C3) )
               => ( groups2240296850493347238T_real(G,A4) = groups2240296850493347238T_real(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4792_sum_Osame__carrierI,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: fun_real_real,H2: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C3)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C3)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C3)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C3,A4))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C3,B4))
                 => ( aa_real_real(H2,B3) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C3) = groups8097168146408367636l_real(H2,C3) )
               => ( groups8097168146408367636l_real(G,A4) = groups8097168146408367636l_real(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4793_sum_Osame__carrierI,axiom,
    ! [C3: set_complex,A4: set_complex,B4: set_complex,G: fun_complex_real,H2: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),C3)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C3)
         => ( ! [A3: complex] :
                ( aa_set_complex_o(member_complex(A3),minus_811609699411566653omplex(C3,A4))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(C3,B4))
                 => ( aa_complex_real(H2,B3) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C3) = groups5808333547571424918x_real(H2,C3) )
               => ( groups5808333547571424918x_real(G,A4) = groups5808333547571424918x_real(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4794_sum_Osame__carrierI,axiom,
    ! [C3: set_Code_integer,A4: set_Code_integer,B4: set_Code_integer,G: fun_Co3960896333048504454r_real,H2: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C3)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),C3)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),C3)
         => ( ! [A3: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A3),minus_2355218937544613996nteger(C3,A4))
               => ( aa_Code_integer_real(G,A3) = zero_zero_real ) )
           => ( ! [B3: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(C3,B4))
                 => ( aa_Code_integer_real(H2,B3) = zero_zero_real ) )
             => ( ( groups1270011288395367621r_real(G,C3) = groups1270011288395367621r_real(H2,C3) )
               => ( groups1270011288395367621r_real(G,A4) = groups1270011288395367621r_real(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4795_sum_Osame__carrierI,axiom,
    ! [C3: set_VEBT_VEBT,A4: set_VEBT_VEBT,B4: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H2: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C3)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),C3)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),C3)
         => ( ! [A3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),minus_5127226145743854075T_VEBT(C3,A4))
               => ( aa_VEBT_VEBT_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(C3,B4))
                 => ( aa_VEBT_VEBT_rat(H2,B3) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,C3) = groups136491112297645522BT_rat(H2,C3) )
               => ( groups136491112297645522BT_rat(G,A4) = groups136491112297645522BT_rat(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4796_sum_Osame__carrierI,axiom,
    ! [C3: set_real,A4: set_real,B4: set_real,G: fun_real_rat,H2: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C3)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),C3)
       => ( aa_set_real_o(ord_less_eq_set_real(B4),C3)
         => ( ! [A3: real] :
                ( aa_set_real_o(member_real(A3),minus_minus_set_real(C3,A4))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( aa_set_real_o(member_real(B3),minus_minus_set_real(C3,B4))
                 => ( aa_real_rat(H2,B3) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C3) = groups1300246762558778688al_rat(H2,C3) )
               => ( groups1300246762558778688al_rat(G,A4) = groups1300246762558778688al_rat(H2,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4797_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_uint32(G,X) = zero_zero_uint32 ) )
         => ( groups8736914816313324502uint32(G,S2) = groups8736914816313324502uint32(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4798_sum_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Cod8401701657498852871uint32(G,X) = zero_zero_uint32 ) )
         => ( groups8847630953604152069uint32(G,S2) = groups8847630953604152069uint32(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4799_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4800_sum_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_real(G,X) = zero_zero_real ) )
         => ( groups1270011288395367621r_real(G,S2) = groups1270011288395367621r_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4801_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4802_sum_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_rat(G,X) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,S2) = groups6602215022474089585er_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4803_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_nat(G,X) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4804_sum_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_nat(G,X) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,S2) = groups7237345082560585321er_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4805_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_int(G,X) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4806_sum_Omono__neutral__left,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_int(G,X) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(G,S2) = groups7234854612051535045er_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4807_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_uint32(G,X) = zero_zero_uint32 ) )
         => ( groups8736914816313324502uint32(G,T2) = groups8736914816313324502uint32(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4808_sum_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Cod8401701657498852871uint32(G,X) = zero_zero_uint32 ) )
         => ( groups8847630953604152069uint32(G,T2) = groups8847630953604152069uint32(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4809_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4810_sum_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_real(G,X) = zero_zero_real ) )
         => ( groups1270011288395367621r_real(G,T2) = groups1270011288395367621r_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4811_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_rat(G,X) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4812_sum_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_rat(G,X) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,T2) = groups6602215022474089585er_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4813_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_nat(G,X) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4814_sum_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_nat(G,X) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,T2) = groups7237345082560585321er_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4815_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_int(G,X) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4816_sum_Omono__neutral__right,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_int(G,X) = zero_zero_int ) )
         => ( groups7234854612051535045er_int(G,T2) = groups7234854612051535045er_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4817_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_VEBT_VEBT,S2: set_VEBT_VEBT,H2: fun_VEBT_VEBT_uint32,G: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T2)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T2,S2))
             => ( aa_VEBT_VEBT_uint32(H2,X) = zero_zero_uint32 ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S2)
               => ( aa_VEBT_VEBT_uint32(G,X) = aa_VEBT_VEBT_uint32(H2,X) ) )
           => ( groups8325533452322294502uint32(G,S2) = groups8325533452322294502uint32(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4818_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H2: fun_real_uint32,G: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T2,S2))
             => ( aa_real_uint32(H2,X) = zero_zero_uint32 ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S2)
               => ( aa_real_uint32(G,X) = aa_real_uint32(H2,X) ) )
           => ( groups5944083974425963860uint32(G,S2) = groups5944083974425963860uint32(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4819_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H2: fun_complex_uint32,G: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_uint32(H2,X) = zero_zero_uint32 ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_uint32(G,X) = aa_complex_uint32(H2,X) ) )
           => ( groups8736914816313324502uint32(G,S2) = groups8736914816313324502uint32(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4820_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,H2: fun_Co1554742298434871494uint32,G: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Cod8401701657498852871uint32(H2,X) = zero_zero_uint32 ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S2)
               => ( aa_Cod8401701657498852871uint32(G,X) = aa_Cod8401701657498852871uint32(H2,X) ) )
           => ( groups8847630953604152069uint32(G,S2) = groups8847630953604152069uint32(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4821_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_VEBT_VEBT,S2: set_VEBT_VEBT,H2: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T2)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T2,S2))
             => ( aa_VEBT_VEBT_real(H2,X) = zero_zero_real ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S2)
               => ( aa_VEBT_VEBT_real(G,X) = aa_VEBT_VEBT_real(H2,X) ) )
           => ( groups2240296850493347238T_real(G,S2) = groups2240296850493347238T_real(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4822_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H2: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T2,S2))
             => ( aa_real_real(H2,X) = zero_zero_real ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S2)
               => ( aa_real_real(G,X) = aa_real_real(H2,X) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4823_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S2: set_complex,H2: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(H2,X) = zero_zero_real ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_real(G,X) = aa_complex_real(H2,X) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4824_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,H2: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_real(H2,X) = zero_zero_real ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S2)
               => ( aa_Code_integer_real(G,X) = aa_Code_integer_real(H2,X) ) )
           => ( groups1270011288395367621r_real(G,S2) = groups1270011288395367621r_real(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4825_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_VEBT_VEBT,S2: set_VEBT_VEBT,H2: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T2)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T2,S2))
             => ( aa_VEBT_VEBT_rat(H2,X) = zero_zero_rat ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S2)
               => ( aa_VEBT_VEBT_rat(G,X) = aa_VEBT_VEBT_rat(H2,X) ) )
           => ( groups136491112297645522BT_rat(G,S2) = groups136491112297645522BT_rat(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4826_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S2: set_real,H2: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T2,S2))
             => ( aa_real_rat(H2,X) = zero_zero_rat ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S2)
               => ( aa_real_rat(G,X) = aa_real_rat(H2,X) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H2,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4827_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,H2: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T2)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T2,S2))
             => ( aa_VEBT_VEBT_uint32(G,X) = zero_zero_uint32 ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S2)
               => ( aa_VEBT_VEBT_uint32(G,X) = aa_VEBT_VEBT_uint32(H2,X) ) )
           => ( groups8325533452322294502uint32(G,T2) = groups8325533452322294502uint32(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4828_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_uint32,H2: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T2,S2))
             => ( aa_real_uint32(G,X) = zero_zero_uint32 ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S2)
               => ( aa_real_uint32(G,X) = aa_real_uint32(H2,X) ) )
           => ( groups5944083974425963860uint32(G,T2) = groups5944083974425963860uint32(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4829_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_uint32,H2: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_uint32(G,X) = zero_zero_uint32 ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_uint32(G,X) = aa_complex_uint32(H2,X) ) )
           => ( groups8736914816313324502uint32(G,T2) = groups8736914816313324502uint32(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4830_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Co1554742298434871494uint32,H2: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Cod8401701657498852871uint32(G,X) = zero_zero_uint32 ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S2)
               => ( aa_Cod8401701657498852871uint32(G,X) = aa_Cod8401701657498852871uint32(H2,X) ) )
           => ( groups8847630953604152069uint32(G,T2) = groups8847630953604152069uint32(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4831_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H2: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T2)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T2,S2))
             => ( aa_VEBT_VEBT_real(G,X) = zero_zero_real ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S2)
               => ( aa_VEBT_VEBT_real(G,X) = aa_VEBT_VEBT_real(H2,X) ) )
           => ( groups2240296850493347238T_real(G,T2) = groups2240296850493347238T_real(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4832_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_real,H2: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T2,S2))
             => ( aa_real_real(G,X) = zero_zero_real ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S2)
               => ( aa_real_real(G,X) = aa_real_real(H2,X) ) )
           => ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4833_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S2: set_complex,G: fun_complex_real,H2: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T2)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T2,S2))
             => ( aa_complex_real(G,X) = zero_zero_real ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_real(G,X) = aa_complex_real(H2,X) ) )
           => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4834_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_Code_integer,S2: set_Code_integer,G: fun_Co3960896333048504454r_real,H2: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S2),T2)
       => ( ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),minus_2355218937544613996nteger(T2,S2))
             => ( aa_Code_integer_real(G,X) = zero_zero_real ) )
         => ( ! [X: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X),S2)
               => ( aa_Code_integer_real(G,X) = aa_Code_integer_real(H2,X) ) )
           => ( groups1270011288395367621r_real(G,T2) = groups1270011288395367621r_real(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4835_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H2: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T2)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S2),T2)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),minus_5127226145743854075T_VEBT(T2,S2))
             => ( aa_VEBT_VEBT_rat(G,X) = zero_zero_rat ) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),S2)
               => ( aa_VEBT_VEBT_rat(G,X) = aa_VEBT_VEBT_rat(H2,X) ) )
           => ( groups136491112297645522BT_rat(G,T2) = groups136491112297645522BT_rat(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4836_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S2: set_real,G: fun_real_rat,H2: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T2)
     => ( aa_set_real_o(ord_less_eq_set_real(S2),T2)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),minus_minus_set_real(T2,S2))
             => ( aa_real_rat(G,X) = zero_zero_rat ) )
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),S2)
               => ( aa_real_rat(G,X) = aa_real_rat(H2,X) ) )
           => ( groups1300246762558778688al_rat(G,T2) = groups1300246762558778688al_rat(H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4837_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5808333547571424918x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,B4))),groups5808333547571424918x_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4838_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups1270011288395367621r_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A4,B4))),groups1270011288395367621r_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4839_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5058264527183730370ex_rat(G,A4) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,B4))),groups5058264527183730370ex_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4840_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups6602215022474089585er_rat(G,A4) = aa_rat_rat(plus_plus_rat(groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A4,B4))),groups6602215022474089585er_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4841_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5693394587270226106ex_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A4,B4))),groups5693394587270226106ex_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4842_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups7237345082560585321er_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A4,B4))),groups7237345082560585321er_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4843_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A4)
       => ( groups5690904116761175830ex_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A4,B4))),groups5690904116761175830ex_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4844_sum_Osubset__diff,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,G: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( groups7234854612051535045er_int(G,A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups7234854612051535045er_int(G,minus_2355218937544613996nteger(A4,B4))),groups7234854612051535045er_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4845_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( groups2906978787729119204at_rat(G,A4) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A4,B4))),groups2906978787729119204at_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4846_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(G),A4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),minus_minus_set_nat(A4,B4))),aa_set_nat_int(groups3539618377306564664at_int(G),B4)) ) ) ) ).

% sum.subset_diff
tff(fact_4847_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups8736914816313324502uint32(F,minus_811609699411566653omplex(A4,B4)) = minus_minus_uint32(groups8736914816313324502uint32(F,A4),groups8736914816313324502uint32(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4848_sum__diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,F: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
       => ( groups8847630953604152069uint32(F,minus_2355218937544613996nteger(A4,B4)) = minus_minus_uint32(groups8847630953604152069uint32(F,A4),groups8847630953604152069uint32(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4849_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A4,B4)) = aa_real_real(minus_minus_real(groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4850_sum__diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
       => ( groups1270011288395367621r_real(F,minus_2355218937544613996nteger(A4,B4)) = aa_real_real(minus_minus_real(groups1270011288395367621r_real(F,A4)),groups1270011288395367621r_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4851_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A4)),groups5690904116761175830ex_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4852_sum__diff,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer,F: fun_Code_integer_int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
       => ( groups7234854612051535045er_int(F,minus_2355218937544613996nteger(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups7234854612051535045er_int(F,A4)),groups7234854612051535045er_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4853_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( groups833757482993574392uint32(F,minus_minus_set_nat(A4,B4)) = minus_minus_uint32(groups833757482993574392uint32(F,A4),groups833757482993574392uint32(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4854_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( aa_set_nat_int(groups3539618377306564664at_int(F),minus_minus_set_nat(A4,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),A4)),aa_set_nat_int(groups3539618377306564664at_int(F),B4)) ) ) ) ).

% sum_diff
tff(fact_4855_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( groups5712668689793887828uint32(F,minus_minus_set_int(A4,B4)) = minus_minus_uint32(groups5712668689793887828uint32(F,A4),groups5712668689793887828uint32(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4856_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A4,B4)) = aa_real_real(minus_minus_real(groups8778361861064173332t_real(F,A4)),groups8778361861064173332t_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_4857_sum__diff__nat,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A4)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_4858_sum__diff__nat,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,F: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
       => ( groups7237345082560585321er_nat(F,minus_2355218937544613996nteger(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups7237345082560585321er_nat(F,A4)),groups7237345082560585321er_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_4859_sum__diff__nat,axiom,
    ! [B4: set_int,A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_4860_sum__diff__nat,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A4,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_4861_sum__diff1__nat,axiom,
    ! [F: fun_VEBT_VEBT_nat,A4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( groups771621172384141258BT_nat(F,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT))) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A4)),aa_VEBT_VEBT_nat(F,A)),groups771621172384141258BT_nat(F,A4)) ) ).

% sum_diff1_nat
tff(fact_4862_sum__diff1__nat,axiom,
    ! [F: fun_real_nat,A4: set_real,A: real] :
      ( groups1935376822645274424al_nat(F,minus_minus_set_real(A4,aa_set_real_set_real(insert_real(A),bot_bot_set_real))) = $ite(aa_set_real_o(member_real(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A4)),aa_real_nat(F,A)),groups1935376822645274424al_nat(F,A4)) ) ).

% sum_diff1_nat
tff(fact_4863_sum__diff1__nat,axiom,
    ! [F: fun_set_nat_nat,A4: set_set_nat,A: set_nat] :
      ( groups8294997508430121362at_nat(F,minus_2163939370556025621et_nat(A4,aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat))) = $ite(aa_set_set_nat_o(member_set_nat(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A4)),aa_set_nat_nat(F,A)),groups8294997508430121362at_nat(F,A4)) ) ).

% sum_diff1_nat
tff(fact_4864_sum__diff1__nat,axiom,
    ! [F: fun_int_nat,A4: set_int,A: int] :
      ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),bot_bot_set_int))) = $ite(aa_set_int_o(member_int(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),aa_int_nat(F,A)),groups4541462559716669496nt_nat(F,A4)) ) ).

% sum_diff1_nat
tff(fact_4865_sum__diff1__nat,axiom,
    ! [F: fun_o_nat,A4: set_o,A: $o] :
      ( groups8507830703676809646_o_nat(F,minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),bot_bot_set_o))) = $ite(aa_set_o_o(member_o((A)),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(F,A4)),aa_o_nat(F,(A))),groups8507830703676809646_o_nat(F,A4)) ) ).

% sum_diff1_nat
tff(fact_4866_sum__diff1__nat,axiom,
    ! [F: fun_nat_nat,A4: set_nat,A: nat] :
      ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat))) = $ite(aa_set_nat_o(member_nat(A),A4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_nat_nat(F,A)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)) ) ).

% sum_diff1_nat
tff(fact_4867_sum__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
       => ( ! [B3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2240296850493347238T_real(F,A4)),groups2240296850493347238T_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4868_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),groups8097168146408367636l_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4869_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A4)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4870_sum__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,F: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [B3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(B4,A4))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,B3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(F,A4)),groups1270011288395367621r_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4871_sum__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
       => ( ! [B3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups136491112297645522BT_rat(F,A4)),groups136491112297645522BT_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4872_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B4)
     => ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
       => ( ! [B3: real] :
              ( aa_set_real_o(member_real(B3),minus_minus_set_real(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A4)),groups1300246762558778688al_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4873_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A4),B4)
       => ( ! [B3: complex] :
              ( aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A4)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4874_sum__mono2,axiom,
    ! [B4: set_Code_integer,A4: set_Code_integer,F: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A4),B4)
       => ( ! [B3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B3),minus_2355218937544613996nteger(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups6602215022474089585er_rat(F,A4)),groups6602215022474089585er_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4875_sum__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
       => ( ! [B3: nat] :
              ( aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B4,A4))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A4)),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4876_sum__mono2,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
       => ( ! [B3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B4,A4))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups771621172384141258BT_nat(F,A4)),groups771621172384141258BT_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_4877_sum_Oremove,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( groups2240296850493347238T_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,Xa)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_4878_sum_Oremove,axiom,
    ! [A4: set_real,Xa: real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => ( groups8097168146408367636l_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,Xa)),groups8097168146408367636l_real(G,minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_4879_sum_Oremove,axiom,
    ! [A4: set_complex,Xa: complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(Xa),A4)
       => ( groups5808333547571424918x_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,Xa)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_4880_sum_Oremove,axiom,
    ! [A4: set_Code_integer,Xa: code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( groups1270011288395367621r_real(G,A4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(G,Xa)),groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_4881_sum_Oremove,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( groups136491112297645522BT_rat(G,A4) = aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(G,Xa)),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_4882_sum_Oremove,axiom,
    ! [A4: set_real,Xa: real,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => ( groups1300246762558778688al_rat(G,A4) = aa_rat_rat(plus_plus_rat(aa_real_rat(G,Xa)),groups1300246762558778688al_rat(G,minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_4883_sum_Oremove,axiom,
    ! [A4: set_complex,Xa: complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_complex_o(member_complex(Xa),A4)
       => ( groups5058264527183730370ex_rat(G,A4) = aa_rat_rat(plus_plus_rat(aa_complex_rat(G,Xa)),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,insert_complex(Xa,bot_bot_set_complex)))) ) ) ) ).

% sum.remove
tff(fact_4884_sum_Oremove,axiom,
    ! [A4: set_Code_integer,Xa: code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( groups6602215022474089585er_rat(G,A4) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(G,Xa)),groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)))) ) ) ) ).

% sum.remove
tff(fact_4885_sum_Oremove,axiom,
    ! [A4: set_VEBT_VEBT,Xa: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => ( groups771621172384141258BT_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,Xa)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A4,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)))) ) ) ) ).

% sum.remove
tff(fact_4886_sum_Oremove,axiom,
    ! [A4: set_real,Xa: real,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => ( groups1935376822645274424al_nat(G,A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(G,Xa)),groups1935376822645274424al_nat(G,minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)))) ) ) ) ).

% sum.remove
tff(fact_4887_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Na: nat,X14: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(X13),I)))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14)),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,X13)))))))) ) ).

% max_idx_list
tff(fact_4888_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)) ).

% cnt_non_neg
tff(fact_4889_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(X13),I))),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Foo),aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,X13))))) ) ).

% height_i_max
tff(fact_4890_Bolzano,axiom,
    ! [A: real,B: real,P2: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A3: real,B3: real,C: real] :
            ( aa_real_o(aa_real_fun_real_o(P2,A3),B3)
           => ( aa_real_o(aa_real_fun_real_o(P2,B3),C)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),B3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),C)
                 => aa_real_o(aa_real_fun_real_o(P2,A3),C) ) ) ) )
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
               => ? [D3: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
                    & ! [A3: real,B3: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),X)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(B3),A3)),D3) )
                       => aa_real_o(aa_real_fun_real_o(P2,A3),B3) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P2,A),B) ) ) ) ).

% Bolzano
tff(fact_4891_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_4892_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_4893_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_4894_option_Osize_I4_J,axiom,
    ! [X2: nat] : ( size_size_option_nat(some_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ) ).

% option.size(4)
tff(fact_4895_max__bot,axiom,
    ! [Xa: set_nat] : ( ord_max_set_nat(bot_bot_set_nat,Xa) = Xa ) ).

% max_bot
tff(fact_4896_max__bot,axiom,
    ! [Xa: assn] : ( ord_max_assn(bot_bot_assn,Xa) = Xa ) ).

% max_bot
tff(fact_4897_max__bot,axiom,
    ! [Xa: set_int] : ( ord_max_set_int(bot_bot_set_int,Xa) = Xa ) ).

% max_bot
tff(fact_4898_max__bot,axiom,
    ! [Xa: set_o] : ( ord_max_set_o(bot_bot_set_o,Xa) = Xa ) ).

% max_bot
tff(fact_4899_max__bot,axiom,
    ! [Xa: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xa) = Xa ) ).

% max_bot
tff(fact_4900_max__bot2,axiom,
    ! [Xa: set_nat] : ( ord_max_set_nat(Xa,bot_bot_set_nat) = Xa ) ).

% max_bot2
tff(fact_4901_max__bot2,axiom,
    ! [Xa: assn] : ( ord_max_assn(Xa,bot_bot_assn) = Xa ) ).

% max_bot2
tff(fact_4902_max__bot2,axiom,
    ! [Xa: set_int] : ( ord_max_set_int(Xa,bot_bot_set_int) = Xa ) ).

% max_bot2
tff(fact_4903_max__bot2,axiom,
    ! [Xa: set_o] : ( ord_max_set_o(Xa,bot_bot_set_o) = Xa ) ).

% max_bot2
tff(fact_4904_max__bot2,axiom,
    ! [Xa: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),bot_bot_nat) = Xa ) ).

% max_bot2
tff(fact_4905_max__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)) ) ).

% max_Suc_Suc
tff(fact_4906_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_4907_max__nat_Oleft__neutral,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A) = A ) ).

% max_nat.left_neutral
tff(fact_4908_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_4909_max__nat_Oright__neutral,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),zero_zero_nat) = A ) ).

% max_nat.right_neutral
tff(fact_4910_max__0L,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Na) = Na ) ).

% max_0L
tff(fact_4911_max__0R,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),zero_zero_nat) = Na ) ).

% max_0R
tff(fact_4912_max__0__1_I2_J,axiom,
    aa_real_real(ord_max_real(one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_4913_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_4914_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_4915_max__0__1_I2_J,axiom,
    aa_Cod1783974652533396728nteger(ord_max_Code_integer(one_one_Code_integer),zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_4916_max__0__1_I2_J,axiom,
    aa_int_int(ord_max_int(one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_4917_max__0__1_I1_J,axiom,
    aa_real_real(ord_max_real(zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_4918_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_4919_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_4920_max__0__1_I1_J,axiom,
    aa_Cod1783974652533396728nteger(ord_max_Code_integer(zero_z3403309356797280102nteger),one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_4921_max__0__1_I1_J,axiom,
    aa_int_int(ord_max_int(zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_4922_Max__insert,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(insert_o((Xa)),A4))
        <=> aa_o_o(aa_o_fun_o_o(ord_max_o,(Xa)),aa_set_o_o(lattic1921953407002678535_Max_o,A4)) ) ) ) ).

% Max_insert
tff(fact_4923_Max__insert,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( lattic4901227151466704046nteger(insert_Code_integer(Xa,A4)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),lattic4901227151466704046nteger(A4)) ) ) ) ).

% Max_insert
tff(fact_4924_Max__insert,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( lattic8263393255366662781ax_int(aa_set_int_set_int(insert_int(Xa),A4)) = aa_int_int(ord_max_int(Xa),lattic8263393255366662781ax_int(A4)) ) ) ) ).

% Max_insert
tff(fact_4925_Max__insert,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(Xa),A4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)) ) ) ) ).

% Max_insert
tff(fact_4926_max__def,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B),B,A) ) ).

% max_def
tff(fact_4927_max__def,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_max_set_int(A,B) = $ite(aa_set_int_o(ord_less_eq_set_int(A),B),B,A) ) ).

% max_def
tff(fact_4928_max__def,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_rat(ord_max_rat(A),B) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B),B,A) ) ).

% max_def
tff(fact_4929_max__def,axiom,
    ! [A: num,B: num] :
      ( aa_num_num(ord_max_num(A),B) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B),B,A) ) ).

% max_def
tff(fact_4930_max__def,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),B,A) ) ).

% max_def
tff(fact_4931_max__def,axiom,
    ! [A: int,B: int] :
      ( aa_int_int(ord_max_int(A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),B,A) ) ).

% max_def
tff(fact_4932_max__absorb1,axiom,
    ! [Y: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Xa)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),Y) = Xa ) ) ).

% max_absorb1
tff(fact_4933_max__absorb1,axiom,
    ! [Y: set_int,Xa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),Xa)
     => ( ord_max_set_int(Xa,Y) = Xa ) ) ).

% max_absorb1
tff(fact_4934_max__absorb1,axiom,
    ! [Y: rat,Xa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xa)
     => ( aa_rat_rat(ord_max_rat(Xa),Y) = Xa ) ) ).

% max_absorb1
tff(fact_4935_max__absorb1,axiom,
    ! [Y: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xa)
     => ( aa_num_num(ord_max_num(Xa),Y) = Xa ) ) ).

% max_absorb1
tff(fact_4936_max__absorb1,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y) = Xa ) ) ).

% max_absorb1
tff(fact_4937_max__absorb1,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa)
     => ( aa_int_int(ord_max_int(Xa),Y) = Xa ) ) ).

% max_absorb1
tff(fact_4938_max__absorb2,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Y)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),Y) = Y ) ) ).

% max_absorb2
tff(fact_4939_max__absorb2,axiom,
    ! [Xa: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xa),Y)
     => ( ord_max_set_int(Xa,Y) = Y ) ) ).

% max_absorb2
tff(fact_4940_max__absorb2,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
     => ( aa_rat_rat(ord_max_rat(Xa),Y) = Y ) ) ).

% max_absorb2
tff(fact_4941_max__absorb2,axiom,
    ! [Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
     => ( aa_num_num(ord_max_num(Xa),Y) = Y ) ) ).

% max_absorb2
tff(fact_4942_max__absorb2,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y) = Y ) ) ).

% max_absorb2
tff(fact_4943_max__absorb2,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => ( aa_int_int(ord_max_int(Xa),Y) = Y ) ) ).

% max_absorb2
tff(fact_4944_max__add__distrib__right,axiom,
    ! [Xa: real,Y: real,Z: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(ord_max_real(Y),Z)) = aa_real_real(ord_max_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Z)) ) ).

% max_add_distrib_right
tff(fact_4945_max__add__distrib__right,axiom,
    ! [Xa: rat,Y: rat,Z: rat] : ( aa_rat_rat(plus_plus_rat(Xa),aa_rat_rat(ord_max_rat(Y),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xa),Y)),aa_rat_rat(plus_plus_rat(Xa),Z)) ) ).

% max_add_distrib_right
tff(fact_4946_max__add__distrib__right,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Z)) ) ).

% max_add_distrib_right
tff(fact_4947_max__add__distrib__right,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] : ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Xa),aa_Cod1783974652533396728nteger(ord_max_Code_integer(Y),Z)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Xa),Y)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Xa),Z)) ) ).

% max_add_distrib_right
tff(fact_4948_max__add__distrib__right,axiom,
    ! [Xa: int,Y: int,Z: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),aa_int_int(ord_max_int(Y),Z)) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Z)) ) ).

% max_add_distrib_right
tff(fact_4949_max__add__distrib__left,axiom,
    ! [Xa: real,Y: real,Z: real] : ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ord_max_real(Xa),Y)),Z) = aa_real_real(ord_max_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ) ).

% max_add_distrib_left
tff(fact_4950_max__add__distrib__left,axiom,
    ! [Xa: rat,Y: rat,Z: rat] : ( aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(Xa),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xa),Z)),aa_rat_rat(plus_plus_rat(Y),Z)) ) ).

% max_add_distrib_left
tff(fact_4951_max__add__distrib__left,axiom,
    ! [Xa: nat,Y: nat,Z: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ) ).

% max_add_distrib_left
tff(fact_4952_max__add__distrib__left,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] : ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),Y)),Z) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Xa),Z)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Y),Z)) ) ).

% max_add_distrib_left
tff(fact_4953_max__add__distrib__left,axiom,
    ! [Xa: int,Y: int,Z: int] : ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ord_max_int(Xa),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ) ).

% max_add_distrib_left
tff(fact_4954_max__diff__distrib__left,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),Y)),Z) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Xa),Z)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_4955_max__diff__distrib__left,axiom,
    ! [Xa: real,Y: real,Z: real] : ( aa_real_real(minus_minus_real(aa_real_real(ord_max_real(Xa),Y)),Z) = aa_real_real(ord_max_real(aa_real_real(minus_minus_real(Xa),Z)),aa_real_real(minus_minus_real(Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_4956_max__diff__distrib__left,axiom,
    ! [Xa: int,Y: int,Z: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ord_max_int(Xa),Y)),Z) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_4957_nat__add__max__right,axiom,
    ! [M: nat,Na: 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,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% nat_add_max_right
tff(fact_4958_nat__add__max__left,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),Q3)) ) ).

% nat_add_max_left
tff(fact_4959_nat__mult__max__right,axiom,
    ! [M: nat,Na: 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,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ) ).

% nat_mult_max_right
tff(fact_4960_nat__mult__max__left,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),Q3)) ) ).

% nat_mult_max_left
tff(fact_4961_nat__minus__add__max,axiom,
    ! [Na: 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,Na),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),M) ) ).

% nat_minus_add_max
tff(fact_4962_Max_Oin__idem,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => ( aa_real_real(ord_max_real(Xa),lattic4275903605611617917x_real(A4)) = lattic4275903605611617917x_real(A4) ) ) ) ).

% Max.in_idem
tff(fact_4963_Max_Oin__idem,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),lattic4901227151466704046nteger(A4)) = lattic4901227151466704046nteger(A4) ) ) ) ).

% Max.in_idem
tff(fact_4964_Max_Oin__idem,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( aa_int_int(ord_max_int(Xa),lattic8263393255366662781ax_int(A4)) = lattic8263393255366662781ax_int(A4) ) ) ) ).

% Max.in_idem
tff(fact_4965_Max_Oin__idem,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)) = aa_set_nat_nat(lattic8265883725875713057ax_nat,A4) ) ) ) ).

% Max.in_idem
tff(fact_4966_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_Leaf((A),(B))) = one_one_real ) ).

% VEBT_internal.cnt.simps(1)
tff(fact_4967_hom__Max__commute,axiom,
    ! [H2: fun_o_o,N6: set_o] :
      ( ! [X: $o,Y3: $o] :
          ( aa_o_o(H2,aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),(Y3)))
        <=> aa_o_o(aa_o_fun_o_o(ord_max_o,aa_o_o(H2,(X))),aa_o_o(H2,(Y3))) )
     => ( aa_set_o_o(finite_finite_o,N6)
       => ( ( N6 != bot_bot_set_o )
         => ( aa_o_o(H2,aa_set_o_o(lattic1921953407002678535_Max_o,N6))
          <=> aa_set_o_o(lattic1921953407002678535_Max_o,image_o_o(H2,N6)) ) ) ) ) ).

% hom_Max_commute
tff(fact_4968_hom__Max__commute,axiom,
    ! [H2: fun_Co8499238673658658103nteger,N6: set_Code_integer] :
      ( ! [X: code_integer,Y3: code_integer] : ( aa_Cod1783974652533396728nteger(H2,aa_Cod1783974652533396728nteger(ord_max_Code_integer(X),Y3)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_Cod1783974652533396728nteger(H2,X)),aa_Cod1783974652533396728nteger(H2,Y3)) )
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,N6)
       => ( ( N6 != bot_bo3990330152332043303nteger )
         => ( aa_Cod1783974652533396728nteger(H2,lattic4901227151466704046nteger(N6)) = lattic4901227151466704046nteger(image_4470545334726330049nteger(H2,N6)) ) ) ) ) ).

% hom_Max_commute
tff(fact_4969_hom__Max__commute,axiom,
    ! [H2: fun_int_int,N6: set_int] :
      ( ! [X: int,Y3: int] : ( aa_int_int(H2,aa_int_int(ord_max_int(X),Y3)) = aa_int_int(ord_max_int(aa_int_int(H2,X)),aa_int_int(H2,Y3)) )
     => ( aa_set_int_o(finite_finite_int,N6)
       => ( ( N6 != bot_bot_set_int )
         => ( aa_int_int(H2,lattic8263393255366662781ax_int(N6)) = lattic8263393255366662781ax_int(aa_set_int_set_int(image_int_int(H2),N6)) ) ) ) ) ).

% hom_Max_commute
tff(fact_4970_hom__Max__commute,axiom,
    ! [H2: fun_nat_nat,N6: set_nat] :
      ( ! [X: nat,Y3: nat] : ( aa_nat_nat(H2,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(H2,X)),aa_nat_nat(H2,Y3)) )
     => ( aa_set_nat_o(finite_finite_nat,N6)
       => ( ( N6 != bot_bot_set_nat )
         => ( aa_nat_nat(H2,aa_set_nat_nat(lattic8265883725875713057ax_nat,N6)) = aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(image_nat_nat(H2),N6)) ) ) ) ) ).

% hom_Max_commute
tff(fact_4971_Max_Osubset,axiom,
    ! [A4: set_o,B4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( B4 != bot_bot_set_o )
       => ( aa_set_o_o(ord_less_eq_set_o(B4),A4)
         => ( aa_o_o(aa_o_fun_o_o(ord_max_o,aa_set_o_o(lattic1921953407002678535_Max_o,B4)),aa_set_o_o(lattic1921953407002678535_Max_o,A4))
          <=> aa_set_o_o(lattic1921953407002678535_Max_o,A4) ) ) ) ) ).

% Max.subset
tff(fact_4972_Max_Osubset,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( B4 != bot_bo3990330152332043303nteger )
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B4),A4)
         => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(lattic4901227151466704046nteger(B4)),lattic4901227151466704046nteger(A4)) = lattic4901227151466704046nteger(A4) ) ) ) ) ).

% Max.subset
tff(fact_4973_Max_Osubset,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( B4 != bot_bot_set_int )
       => ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
         => ( aa_int_int(ord_max_int(lattic8263393255366662781ax_int(B4)),lattic8263393255366662781ax_int(A4)) = lattic8263393255366662781ax_int(A4) ) ) ) ) ).

% Max.subset
tff(fact_4974_Max_Osubset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( B4 != bot_bot_set_nat )
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,B4)),aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)) = aa_set_nat_nat(lattic8265883725875713057ax_nat,A4) ) ) ) ) ).

% Max.subset
tff(fact_4975_Max_Oclosed,axiom,
    ! [A4: set_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X: real,Y3: real] : aa_set_real_o(member_real(aa_real_real(ord_max_real(X),Y3)),aa_set_real_set_real(insert_real(X),aa_set_real_set_real(insert_real(Y3),bot_bot_set_real)))
         => aa_set_real_o(member_real(lattic4275903605611617917x_real(A4)),A4) ) ) ) ).

% Max.closed
tff(fact_4976_Max_Oclosed,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X: $o,Y3: $o] : aa_set_o_o(member_o(aa_o_o(aa_o_fun_o_o(ord_max_o,(X)),(Y3))),aa_set_o_set_o(insert_o((X)),aa_set_o_set_o(insert_o((Y3)),bot_bot_set_o)))
         => aa_set_o_o(member_o(aa_set_o_o(lattic1921953407002678535_Max_o,A4)),A4) ) ) ) ).

% Max.closed
tff(fact_4977_Max_Oclosed,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X: code_integer,Y3: code_integer] : aa_set5643387332196575111eger_o(member_Code_integer(aa_Cod1783974652533396728nteger(ord_max_Code_integer(X),Y3)),insert_Code_integer(X,insert_Code_integer(Y3,bot_bo3990330152332043303nteger)))
         => aa_set5643387332196575111eger_o(member_Code_integer(lattic4901227151466704046nteger(A4)),A4) ) ) ) ).

% Max.closed
tff(fact_4978_Max_Oclosed,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X: int,Y3: int] : aa_set_int_o(member_int(aa_int_int(ord_max_int(X),Y3)),aa_set_int_set_int(insert_int(X),aa_set_int_set_int(insert_int(Y3),bot_bot_set_int)))
         => aa_set_int_o(member_int(lattic8263393255366662781ax_int(A4)),A4) ) ) ) ).

% Max.closed
tff(fact_4979_Max_Oclosed,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X: nat,Y3: nat] : aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y3)),aa_set_nat_set_nat(insert_nat(X),aa_set_nat_set_nat(insert_nat(Y3),bot_bot_set_nat)))
         => aa_set_nat_o(member_nat(aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)),A4) ) ) ) ).

% Max.closed
tff(fact_4980_Max_Oinsert__not__elem,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ~ aa_set_real_o(member_real(Xa),A4)
       => ( ( A4 != bot_bot_set_real )
         => ( lattic4275903605611617917x_real(aa_set_real_set_real(insert_real(Xa),A4)) = aa_real_real(ord_max_real(Xa),lattic4275903605611617917x_real(A4)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4981_Max_Oinsert__not__elem,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ~ aa_set_o_o(member_o((Xa)),A4)
       => ( ( A4 != bot_bot_set_o )
         => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(insert_o((Xa)),A4))
          <=> aa_o_o(aa_o_fun_o_o(ord_max_o,(Xa)),aa_set_o_o(lattic1921953407002678535_Max_o,A4)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4982_Max_Oinsert__not__elem,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( ( A4 != bot_bo3990330152332043303nteger )
         => ( lattic4901227151466704046nteger(insert_Code_integer(Xa,A4)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),lattic4901227151466704046nteger(A4)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4983_Max_Oinsert__not__elem,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ~ aa_set_int_o(member_int(Xa),A4)
       => ( ( A4 != bot_bot_set_int )
         => ( lattic8263393255366662781ax_int(aa_set_int_set_int(insert_int(Xa),A4)) = aa_int_int(ord_max_int(Xa),lattic8263393255366662781ax_int(A4)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4984_Max_Oinsert__not__elem,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(Xa),A4)
       => ( ( A4 != bot_bot_set_nat )
         => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(Xa),A4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),aa_set_nat_nat(lattic8265883725875713057ax_nat,A4)) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4985_Max_Oinsert__remove,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(lattic1921953407002678535_Max_o,aa_set_o_set_o(insert_o((Xa)),A4))
      <=> $ite(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)) = bot_bot_set_o,(Xa),aa_o_o(aa_o_fun_o_o(ord_max_o,(Xa)),aa_set_o_o(lattic1921953407002678535_Max_o,minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))))) ) ) ).

% Max.insert_remove
tff(fact_4986_Max_Oinsert__remove,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( lattic4901227151466704046nteger(insert_Code_integer(Xa,A4)) = $ite(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)) = bot_bo3990330152332043303nteger,Xa,aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),lattic4901227151466704046nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger))))) ) ) ).

% Max.insert_remove
tff(fact_4987_Max_Oinsert__remove,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( lattic8263393255366662781ax_int(aa_set_int_set_int(insert_int(Xa),A4)) = $ite(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) = bot_bot_set_int,Xa,aa_int_int(ord_max_int(Xa),lattic8263393255366662781ax_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))))) ) ) ).

% Max.insert_remove
tff(fact_4988_Max_Oinsert__remove,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(insert_nat(Xa),A4)) = $ite(minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) = bot_bot_set_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),aa_set_nat_nat(lattic8265883725875713057ax_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))))) ) ) ).

% Max.insert_remove
tff(fact_4989_Max_Oremove,axiom,
    ! [A4: set_real,Xa: real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => ( lattic4275903605611617917x_real(A4) = $ite(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)) = bot_bot_set_real,Xa,aa_real_real(ord_max_real(Xa),lattic4275903605611617917x_real(minus_minus_set_real(A4,aa_set_real_set_real(insert_real(Xa),bot_bot_set_real))))) ) ) ) ).

% Max.remove
tff(fact_4990_Max_Oremove,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_set_o_o(lattic1921953407002678535_Max_o,A4)
        <=> $ite(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)) = bot_bot_set_o,(Xa),aa_o_o(aa_o_fun_o_o(ord_max_o,(Xa)),aa_set_o_o(lattic1921953407002678535_Max_o,minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))))) ) ) ) ).

% Max.remove
tff(fact_4991_Max_Oremove,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A4)
       => ( lattic4901227151466704046nteger(A4) = $ite(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger)) = bot_bo3990330152332043303nteger,Xa,aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),lattic4901227151466704046nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger))))) ) ) ) ).

% Max.remove
tff(fact_4992_Max_Oremove,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( lattic8263393255366662781ax_int(A4) = $ite(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) = bot_bot_set_int,Xa,aa_int_int(ord_max_int(Xa),lattic8263393255366662781ax_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))))) ) ) ) ).

% Max.remove
tff(fact_4993_Max_Oremove,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,A4) = $ite(minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) = bot_bot_set_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),aa_set_nat_nat(lattic8265883725875713057ax_nat,minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))))) ) ) ) ).

% Max.remove
tff(fact_4994_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) = A ) ) ).

% max.absorb3
tff(fact_4995_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(ord_max_real(A),B) = A ) ) ).

% max.absorb3
tff(fact_4996_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb3
tff(fact_4997_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb3
tff(fact_4998_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_4999_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb3
tff(fact_5000_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) = B ) ) ).

% max.absorb4
tff(fact_5001_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(ord_max_real(A),B) = B ) ) ).

% max.absorb4
tff(fact_5002_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb4
tff(fact_5003_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb4
tff(fact_5004_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_5005_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb4
tff(fact_5006_max__less__iff__conj,axiom,
    ! [Xa: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),Y)),Z)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Z)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5007_max__less__iff__conj,axiom,
    ! [Xa: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(Xa),Y)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5008_max__less__iff__conj,axiom,
    ! [Xa: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(Xa),Y)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5009_max__less__iff__conj,axiom,
    ! [Xa: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(ord_max_num(Xa),Y)),Z)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),Z)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5010_max__less__iff__conj,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5011_max__less__iff__conj,axiom,
    ! [Xa: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(Xa),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5012_max_Obounded__iff,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(B),C2)),A)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A) ) ) ).

% max.bounded_iff
tff(fact_5013_max_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C2)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% max.bounded_iff
tff(fact_5014_max_Obounded__iff,axiom,
    ! [B: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B),C2)),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% max.bounded_iff
tff(fact_5015_max_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.bounded_iff
tff(fact_5016_max_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B),C2)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.bounded_iff
tff(fact_5017_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) = B ) ) ).

% max.absorb2
tff(fact_5018_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb2
tff(fact_5019_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb2
tff(fact_5020_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_5021_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb2
tff(fact_5022_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) = A ) ) ).

% max.absorb1
tff(fact_5023_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb1
tff(fact_5024_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb1
tff(fact_5025_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_5026_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb1
tff(fact_5027_option_Osize__gen_I2_J,axiom,
    ! [Xa: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : ( size_o8335143837870341156at_nat(Xa,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(Xa,X2)),aa_nat_nat(suc,zero_zero_nat)) ) ).

% option.size_gen(2)
tff(fact_5028_option_Osize__gen_I2_J,axiom,
    ! [Xa: fun_nat_nat,X2: nat] : ( size_option_nat(Xa,some_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Xa,X2)),aa_nat_nat(suc,zero_zero_nat)) ) ).

% option.size_gen(2)
tff(fact_5029_option_Osize__gen_I1_J,axiom,
    ! [Xa: fun_Pr2602117396471556196at_nat] : ( size_o8335143837870341156at_nat(Xa,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ) ).

% option.size_gen(1)
tff(fact_5030_option_Osize__gen_I1_J,axiom,
    ! [Xa: fun_nat_nat] : ( size_option_nat(Xa,none_nat) = aa_nat_nat(suc,zero_zero_nat) ) ).

% option.size_gen(1)
tff(fact_5031_max_Omono,axiom,
    ! [C2: code_integer,A: code_integer,D: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(C2),D)),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B)) ) ) ).

% max.mono
tff(fact_5032_max_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C2),D)),aa_rat_rat(ord_max_rat(A),B)) ) ) ).

% max.mono
tff(fact_5033_max_Omono,axiom,
    ! [C2: num,A: num,D: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(C2),D)),aa_num_num(ord_max_num(A),B)) ) ) ).

% max.mono
tff(fact_5034_max_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C2),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_5035_max_Omono,axiom,
    ! [C2: int,A: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(C2),D)),aa_int_int(ord_max_int(A),B)) ) ) ).

% max.mono
tff(fact_5036_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( A = aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) ) ) ).

% max.orderE
tff(fact_5037_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.orderE
tff(fact_5038_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.orderE
tff(fact_5039_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_5040_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(ord_max_int(A),B) ) ) ).

% max.orderE
tff(fact_5041_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ).

% max.orderI
tff(fact_5042_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(ord_max_rat(A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% max.orderI
tff(fact_5043_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(ord_max_num(A),B) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% max.orderI
tff(fact_5044_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_5045_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(ord_max_int(A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_5046_max_OboundedE,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(B),C2)),A)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A) ) ) ).

% max.boundedE
tff(fact_5047_max_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C2)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A) ) ) ).

% max.boundedE
tff(fact_5048_max_OboundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B),C2)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A) ) ) ).

% max.boundedE
tff(fact_5049_max_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A) ) ) ).

% max.boundedE
tff(fact_5050_max_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A) ) ) ).

% max.boundedE
tff(fact_5051_max_OboundedI,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(B),C2)),A) ) ) ).

% max.boundedI
tff(fact_5052_max_OboundedI,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C2)),A) ) ) ).

% max.boundedI
tff(fact_5053_max_OboundedI,axiom,
    ! [B: num,A: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B),C2)),A) ) ) ).

% max.boundedI
tff(fact_5054_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A) ) ) ).

% max.boundedI
tff(fact_5055_max_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B),C2)),A) ) ) ).

% max.boundedI
tff(fact_5056_max_Oorder__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
    <=> ( A = aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) ) ) ).

% max.order_iff
tff(fact_5057_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.order_iff
tff(fact_5058_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.order_iff
tff(fact_5059_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_5060_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(ord_max_int(A),B) ) ) ).

% max.order_iff
tff(fact_5061_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B)) ).

% max.cobounded1
tff(fact_5062_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(ord_max_rat(A),B)) ).

% max.cobounded1
tff(fact_5063_max_Ocobounded1,axiom,
    ! [A: num,B: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(ord_max_num(A),B)) ).

% max.cobounded1
tff(fact_5064_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_5065_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(ord_max_int(A),B)) ).

% max.cobounded1
tff(fact_5066_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B)) ).

% max.cobounded2
tff(fact_5067_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(A),B)) ).

% max.cobounded2
tff(fact_5068_max_Ocobounded2,axiom,
    ! [B: num,A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),aa_num_num(ord_max_num(A),B)) ).

% max.cobounded2
tff(fact_5069_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_5070_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(ord_max_int(A),B)) ).

% max.cobounded2
tff(fact_5071_le__max__iff__disj,axiom,
    ! [Z: code_integer,Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),Y))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Xa)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_5072_le__max__iff__disj,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(Xa),Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xa)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_5073_le__max__iff__disj,axiom,
    ! [Z: num,Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(ord_max_num(Xa),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xa)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_5074_le__max__iff__disj,axiom,
    ! [Z: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xa)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_5075_le__max__iff__disj,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(ord_max_int(Xa),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xa)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_5076_max_Oabsorb__iff1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
    <=> ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_5077_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_5078_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_5079_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_5080_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(ord_max_int(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_5081_max_Oabsorb__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
    <=> ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_5082_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_5083_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_5084_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_5085_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(ord_max_int(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_5086_max_OcoboundedI1,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B)) ) ).

% max.coboundedI1
tff(fact_5087_max_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.coboundedI1
tff(fact_5088_max_OcoboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),aa_num_num(ord_max_num(A),B)) ) ).

% max.coboundedI1
tff(fact_5089_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_5090_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(ord_max_int(A),B)) ) ).

% max.coboundedI1
tff(fact_5091_max_OcoboundedI2,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B)) ) ).

% max.coboundedI2
tff(fact_5092_max_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.coboundedI2
tff(fact_5093_max_OcoboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),aa_num_num(ord_max_num(A),B)) ) ).

% max.coboundedI2
tff(fact_5094_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_5095_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(ord_max_int(A),B)) ) ).

% max.coboundedI2
tff(fact_5096_max_Ostrict__coboundedI2,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5097_max_Ostrict__coboundedI2,axiom,
    ! [C2: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(ord_max_real(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5098_max_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5099_max_Ostrict__coboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),aa_num_num(ord_max_num(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5100_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5101_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(ord_max_int(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5102_max_Ostrict__coboundedI1,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5103_max_Ostrict__coboundedI1,axiom,
    ! [C2: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(ord_max_real(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5104_max_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5105_max_Ostrict__coboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),aa_num_num(ord_max_num(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5106_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5107_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(ord_max_int(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5108_max_Ostrict__order__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
    <=> ( ( A = aa_Cod1783974652533396728nteger(ord_max_Code_integer(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5109_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(ord_max_real(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5110_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = aa_rat_rat(ord_max_rat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5111_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( ( A = aa_num_num(ord_max_num(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5112_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5113_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(ord_max_int(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5114_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(B),C2)),A)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_5115_max_Ostrict__boundedE,axiom,
    ! [B: real,C2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(B),C2)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_5116_max_Ostrict__boundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(B),C2)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_5117_max_Ostrict__boundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(ord_max_num(B),C2)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_5118_max_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C2)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_5119_max_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(B),C2)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A) ) ) ).

% max.strict_boundedE
tff(fact_5120_less__max__iff__disj,axiom,
    ! [Z: code_integer,Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xa),Y))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Xa)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5121_less__max__iff__disj,axiom,
    ! [Z: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(ord_max_real(Xa),Y))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xa)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5122_less__max__iff__disj,axiom,
    ! [Z: rat,Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(ord_max_rat(Xa),Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xa)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5123_less__max__iff__disj,axiom,
    ! [Z: num,Xa: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(ord_max_num(Xa),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Xa)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5124_less__max__iff__disj,axiom,
    ! [Z: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xa)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5125_less__max__iff__disj,axiom,
    ! [Z: int,Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(ord_max_int(Xa),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xa)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5126_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(L))
         => ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),L)
           => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(L),I)),(Xa))
             => ( aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa)),aa_nat_o(nth_o(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(L),K2)),(Xa))
                         => ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa)),aa_nat_o(nth_o(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5127_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(L))
         => ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),L)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(nth_real(L),I)),Xa)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(nth_real(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(nth_real(L),K2)),Xa)
                         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(nth_real(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5128_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_rat,Xa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_rat(L))
         => ( sorted_wrt_rat(ord_less_eq_rat,L)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(L),I)),Xa)
             => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_nat_rat(nth_rat(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(L),K2)),Xa)
                         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_nat_rat(nth_rat(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5129_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_num,Xa: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_num(L))
         => ( sorted_wrt_num(ord_less_eq_num,L)
           => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(L),I)),Xa)
             => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),aa_nat_num(nth_num(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(L),K2)),Xa)
                         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),aa_nat_num(nth_num(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5130_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L))
         => ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),L)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(L),I)),Xa)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(nth_nat(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(L),K2)),Xa)
                         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(nth_nat(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5131_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(L))
         => ( sorted_wrt_int(ord_less_eq_int,L)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(L),I)),Xa)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(nth_int(L),J))
               => ~ ! [K2: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2)
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),J)
                       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(L),K2)),Xa)
                         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(nth_int(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_5132_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBTi,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_Pr3244165891152107629_VEBTi(enumerate_VEBT_VEBTi(Na,Xs),M) = produc2649746096677893406_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_5133_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_int(Xs))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(Na,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_5134_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(Na,Xs),M) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_5135_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_real,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_real(Xs))
     => ( nth_Pr7767817059697521252t_real(enumerate_real(Na,Xs),M) = produc7837566107596912789t_real(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M),aa_nat_real(nth_real(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_5136_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_o(Xs))
     => ( nth_Pr112076138515278198_nat_o(enumerate_o(Na,Xs),M) = product_Pair_nat_o(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M),aa_nat_o(nth_o(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_5137_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(Na,Xs),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),aa_nat_nat(nth_nat(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_5138_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE5577048379444288083T_assn,Xsi: list_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2497118539674116125T_VEBT(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5139_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE1225457145571040039t_assn,Xsi: list_nat] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2809031099982602151Ti_nat(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5140_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE2561185598974572172i_assn,Xsi: list_VEBT_VEBTi] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L886525131989349516_VEBTi(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5141_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE5641251438918192003t_assn,Xsi: list_int] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2806540629473551875Ti_int(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5142_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: fun_in7755481285845205148T_assn,Xsi: list_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( vEBT_L2018189785592951398T_VEBT(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2018189785592951398T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5143_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: fun_int_fun_nat_assn,Xsi: list_nat] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( vEBT_L8891422820522952478nt_nat(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8891422820522952478nt_nat(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5144_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: fun_in1903134756389476099i_assn,Xsi: list_VEBT_VEBTi] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( vEBT_L114188773329725699_VEBTi(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L114188773329725699_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5145_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: fun_int_fun_int_assn,Xsi: list_int] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( vEBT_L8888932350013902202nt_int(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8888932350013902202nt_int(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5146_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L3204528365124325536T_VEBT(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5147_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L8650695023172932196BT_nat(I5,A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5148_nth__step__trancl,axiom,
    ! [Xs: list_VEBT_VEBTi,R2: set_Pr2227491710730465451_VEBTi,Na: nat,M: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s7982070591426661849_VEBTi(Xs)),one_one_nat))
         => member660371905731732212_VEBTi(produc436343169921013763_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_nat(suc,N)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N)),R2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
         => member660371905731732212_VEBTi(produc436343169921013763_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Na),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),M)),transi2803566869205510612_VEBTi(R2)) ) ) ) ).

% nth_step_trancl
tff(fact_5149_nth__step__trancl,axiom,
    ! [Xs: list_uint32,R2: set_Pr1773385645901665561uint32,Na: nat,M: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s4844771616002835472uint32(Xs)),one_one_nat))
         => member8027108493173000802uint32(produc1400373151660368625uint32(aa_nat_uint32(nth_uint32(Xs),aa_nat_nat(suc,N)),aa_nat_uint32(nth_uint32(Xs),N)),R2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s4844771616002835472uint32(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
         => member8027108493173000802uint32(produc1400373151660368625uint32(aa_nat_uint32(nth_uint32(Xs),Na),aa_nat_uint32(nth_uint32(Xs),M)),transi3114468042090999947uint32(R2)) ) ) ) ).

% nth_step_trancl
tff(fact_5150_nth__step__trancl,axiom,
    ! [Xs: list_int,R2: set_Pr958786334691620121nt_int,Na: nat,M: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat))
         => member5262025264175285858nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(suc,N)),aa_nat_int(nth_int(Xs),N)),R2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
         => member5262025264175285858nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),Na),aa_nat_int(nth_int(Xs),M)),transi6261509568448316235cl_int(R2)) ) ) ) ).

% nth_step_trancl
tff(fact_5151_nth__step__trancl,axiom,
    ! [Xs: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT,Na: nat,M: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat))
         => member568628332442017744T_VEBT(produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(suc,N)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N)),R2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
         => member568628332442017744T_VEBT(produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Na),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),M)),transi8906537157094044885T_VEBT(R2)) ) ) ) ).

% nth_step_trancl
tff(fact_5152_nth__step__trancl,axiom,
    ! [Xs: list_real,R2: set_Pr6218003697084177305l_real,Na: nat,M: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_real(Xs)),one_one_nat))
         => member7849222048561428706l_real(produc4511245868158468465l_real(aa_nat_real(nth_real(Xs),aa_nat_nat(suc,N)),aa_nat_real(nth_real(Xs),N)),R2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
         => member7849222048561428706l_real(produc4511245868158468465l_real(aa_nat_real(nth_real(Xs),Na),aa_nat_real(nth_real(Xs),M)),transi1789104906590519371l_real(R2)) ) ) ) ).

% nth_step_trancl
tff(fact_5153_nth__step__trancl,axiom,
    ! [Xs: list_o,R2: set_Product_prod_o_o,Na: nat,M: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_o(Xs)),one_one_nat))
         => member7466972457876170832od_o_o(product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(suc,N)),aa_nat_o(nth_o(Xs),N)),R2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
         => member7466972457876170832od_o_o(product_Pair_o_o(aa_nat_o(nth_o(Xs),Na),aa_nat_o(nth_o(Xs),M)),transitive_trancl_o(R2)) ) ) ) ).

% nth_step_trancl
tff(fact_5154_nth__step__trancl,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat,Na: nat,M: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat))
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,N))),aa_nat_nat(nth_nat(Xs),N)),R2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),Na)),aa_nat_nat(nth_nat(Xs),M)),transi6264000038957366511cl_nat(R2)) ) ) ) ).

% nth_step_trancl
tff(fact_5155_nth__Cons__pos,axiom,
    ! [Na: nat,Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Xa,Xs)),Na) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_5156_nth__Cons__pos,axiom,
    ! [Na: nat,Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Xa,Xs)),Na) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_5157_nth__Cons__pos,axiom,
    ! [Na: nat,Xa: int,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(nth_int(cons_int(Xa,Xs)),Na) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_5158_nth__Cons__pos,axiom,
    ! [Na: nat,Xa: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(nth_nat(cons_nat(Xa,Xs)),Na) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_5159_nth__Cons__pos,axiom,
    ! [Na: nat,Xa: $o,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(nth_o(cons_o((Xa),Xs)),Na)
      <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_5160_length__nth__simps_I4_J,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat] : ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Xa,Xs)),aa_nat_nat(suc,Na)) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Na) ) ).

% length_nth_simps(4)
tff(fact_5161_length__nth__simps_I4_J,axiom,
    ! [Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi,Na: nat] : ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Xa,Xs)),aa_nat_nat(suc,Na)) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Na) ) ).

% length_nth_simps(4)
tff(fact_5162_length__nth__simps_I4_J,axiom,
    ! [Xa: int,Xs: list_int,Na: nat] : ( aa_nat_int(nth_int(cons_int(Xa,Xs)),aa_nat_nat(suc,Na)) = aa_nat_int(nth_int(Xs),Na) ) ).

% length_nth_simps(4)
tff(fact_5163_length__nth__simps_I4_J,axiom,
    ! [Xa: nat,Xs: list_nat,Na: nat] : ( aa_nat_nat(nth_nat(cons_nat(Xa,Xs)),aa_nat_nat(suc,Na)) = aa_nat_nat(nth_nat(Xs),Na) ) ).

% length_nth_simps(4)
tff(fact_5164_length__nth__simps_I4_J,axiom,
    ! [Xa: $o,Xs: list_o,Na: nat] :
      ( aa_nat_o(nth_o(cons_o((Xa),Xs)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(nth_o(Xs),Na) ) ).

% length_nth_simps(4)
tff(fact_5165_nth__Cons__Suc,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat] : ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Xa,Xs)),aa_nat_nat(suc,Na)) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Na) ) ).

% nth_Cons_Suc
tff(fact_5166_nth__Cons__Suc,axiom,
    ! [Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi,Na: nat] : ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Xa,Xs)),aa_nat_nat(suc,Na)) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Na) ) ).

% nth_Cons_Suc
tff(fact_5167_nth__Cons__Suc,axiom,
    ! [Xa: int,Xs: list_int,Na: nat] : ( aa_nat_int(nth_int(cons_int(Xa,Xs)),aa_nat_nat(suc,Na)) = aa_nat_int(nth_int(Xs),Na) ) ).

% nth_Cons_Suc
tff(fact_5168_nth__Cons__Suc,axiom,
    ! [Xa: nat,Xs: list_nat,Na: nat] : ( aa_nat_nat(nth_nat(cons_nat(Xa,Xs)),aa_nat_nat(suc,Na)) = aa_nat_nat(nth_nat(Xs),Na) ) ).

% nth_Cons_Suc
tff(fact_5169_nth__Cons__Suc,axiom,
    ! [Xa: $o,Xs: list_o,Na: nat] :
      ( aa_nat_o(nth_o(cons_o((Xa),Xs)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(nth_o(Xs),Na) ) ).

% nth_Cons_Suc
tff(fact_5170_length__nth__simps_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Xa,Xs)),zero_zero_nat) = Xa ) ).

% length_nth_simps(3)
tff(fact_5171_length__nth__simps_I3_J,axiom,
    ! [Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi] : ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Xa,Xs)),zero_zero_nat) = Xa ) ).

% length_nth_simps(3)
tff(fact_5172_length__nth__simps_I3_J,axiom,
    ! [Xa: int,Xs: list_int] : ( aa_nat_int(nth_int(cons_int(Xa,Xs)),zero_zero_nat) = Xa ) ).

% length_nth_simps(3)
tff(fact_5173_length__nth__simps_I3_J,axiom,
    ! [Xa: nat,Xs: list_nat] : ( aa_nat_nat(nth_nat(cons_nat(Xa,Xs)),zero_zero_nat) = Xa ) ).

% length_nth_simps(3)
tff(fact_5174_length__nth__simps_I3_J,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_nat_o(nth_o(cons_o((Xa),Xs)),zero_zero_nat)
    <=> (Xa) ) ).

% length_nth_simps(3)
tff(fact_5175_nth__Cons__0,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Xa,Xs)),zero_zero_nat) = Xa ) ).

% nth_Cons_0
tff(fact_5176_nth__Cons__0,axiom,
    ! [Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi] : ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Xa,Xs)),zero_zero_nat) = Xa ) ).

% nth_Cons_0
tff(fact_5177_nth__Cons__0,axiom,
    ! [Xa: int,Xs: list_int] : ( aa_nat_int(nth_int(cons_int(Xa,Xs)),zero_zero_nat) = Xa ) ).

% nth_Cons_0
tff(fact_5178_nth__Cons__0,axiom,
    ! [Xa: nat,Xs: list_nat] : ( aa_nat_nat(nth_nat(cons_nat(Xa,Xs)),zero_zero_nat) = Xa ) ).

% nth_Cons_0
tff(fact_5179_nth__Cons__0,axiom,
    ! [Xa: $o,Xs: list_o] :
      ( aa_nat_o(nth_o(cons_o((Xa),Xs)),zero_zero_nat)
    <=> (Xa) ) ).

% nth_Cons_0
tff(fact_5180_trancl__single,axiom,
    ! [A: uint32,B: uint32] : ( transi3114468042090999947uint32(insert4454361187789264009uint32(produc1400373151660368625uint32(A,B),bot_bo8438649754162204037uint32)) = insert4454361187789264009uint32(produc1400373151660368625uint32(A,B),bot_bo8438649754162204037uint32) ) ).

% trancl_single
tff(fact_5181_trancl__single,axiom,
    ! [A: nat,B: nat] : ( transi6264000038957366511cl_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),bot_bo2099793752762293965at_nat) ) ).

% trancl_single
tff(fact_5182_trancl__single,axiom,
    ! [A: int,B: int] : ( transi6261509568448316235cl_int(insert5033312907999012233nt_int(product_Pair_int_int(A,B),bot_bo1796632182523588997nt_int)) = insert5033312907999012233nt_int(product_Pair_int_int(A,B),bot_bo1796632182523588997nt_int) ) ).

% trancl_single
tff(fact_5183_listI__assn__finite,axiom,
    ! [I5: set_nat,A4: fun_VE905575822219544865i_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] :
      ( ~ aa_set_nat_o(finite_finite_nat,I5)
     => ( vEBT_L1528199826722428489_VEBTi(I5,A4,Xs,Xsi) = bot_bot_assn ) ) ).

% listI_assn_finite
tff(fact_5184_enumerate__simps_I2_J,axiom,
    ! [Na: nat,Xa: int,Xs: list_int] : ( enumerate_int(Na,cons_int(Xa,Xs)) = cons_P2335045147070616083at_int(product_Pair_nat_int(Na,Xa),enumerate_int(aa_nat_nat(suc,Na),Xs)) ) ).

% enumerate_simps(2)
tff(fact_5185_enumerate__simps_I2_J,axiom,
    ! [Na: nat,Xa: $o,Xs: list_o] : ( enumerate_o(Na,cons_o((Xa),Xs)) = cons_P9142372351690779143_nat_o(product_Pair_nat_o(Na,(Xa)),enumerate_o(aa_nat_nat(suc,Na),Xs)) ) ).

% enumerate_simps(2)
tff(fact_5186_enumerate__simps_I2_J,axiom,
    ! [Na: nat,Xa: nat,Xs: list_nat] : ( enumerate_nat(Na,cons_nat(Xa,Xs)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Na),Xa),enumerate_nat(aa_nat_nat(suc,Na),Xs)) ) ).

% enumerate_simps(2)
tff(fact_5187_list__tail__coinc,axiom,
    ! [N1: int,R1: list_int,N22: int,R22: list_int] :
      ( ( cons_int(N1,R1) = cons_int(N22,R22) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
tff(fact_5188_list__tail__coinc,axiom,
    ! [N1: nat,R1: list_nat,N22: nat,R22: list_nat] :
      ( ( cons_nat(N1,R1) = cons_nat(N22,R22) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
tff(fact_5189_list__tail__coinc,axiom,
    ! [N1: $o,R1: list_o,N22: $o,R22: list_o] :
      ( ( cons_o((N1),R1) = cons_o((N22),R22) )
     => ( ( (N1)
        <=> (N22) )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
tff(fact_5190_sorted2,axiom,
    ! [Xa: $o,Y: $o,Zs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),cons_o((Xa),cons_o((Y),Zs)))
    <=> ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(Y))
        & aa_list_o_o(sorted_wrt_o(ord_less_eq_o),cons_o((Y),Zs)) ) ) ).

% sorted2
tff(fact_5191_sorted2,axiom,
    ! [Xa: rat,Y: rat,Zs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,cons_rat(Xa,cons_rat(Y,Zs)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
        & sorted_wrt_rat(ord_less_eq_rat,cons_rat(Y,Zs)) ) ) ).

% sorted2
tff(fact_5192_sorted2,axiom,
    ! [Xa: num,Y: num,Zs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,cons_num(Xa,cons_num(Y,Zs)))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y)
        & sorted_wrt_num(ord_less_eq_num,cons_num(Y,Zs)) ) ) ).

% sorted2
tff(fact_5193_sorted2,axiom,
    ! [Xa: nat,Y: nat,Zs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),cons_nat(Xa,cons_nat(Y,Zs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y)
        & aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),cons_nat(Y,Zs)) ) ) ).

% sorted2
tff(fact_5194_sorted2,axiom,
    ! [Xa: int,Y: int,Zs: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,cons_int(Xa,cons_int(Y,Zs)))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
        & sorted_wrt_int(ord_less_eq_int,cons_int(Y,Zs)) ) ) ).

% sorted2
tff(fact_5195_sorted__simps_I2_J,axiom,
    ! [Xa: real,Ys: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),cons_real(Xa,Ys))
    <=> ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),aa_lis3769085826154824284t_real(set_real2,Ys))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X5) )
        & aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Ys) ) ) ).

% sorted_simps(2)
tff(fact_5196_sorted__simps_I2_J,axiom,
    ! [Xa: $o,Ys: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),cons_o((Xa),Ys))
    <=> ( ! [X5: $o] :
            ( aa_set_o_o(member_o((X5)),aa_list_o_set_o(set_o2,Ys))
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(Xa)),(X5)) )
        & aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Ys) ) ) ).

% sorted_simps(2)
tff(fact_5197_sorted__simps_I2_J,axiom,
    ! [Xa: rat,Ys: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,cons_rat(Xa,Ys))
    <=> ( ! [X5: rat] :
            ( aa_set_rat_o(member_rat(X5),set_rat2(Ys))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X5) )
        & sorted_wrt_rat(ord_less_eq_rat,Ys) ) ) ).

% sorted_simps(2)
tff(fact_5198_sorted__simps_I2_J,axiom,
    ! [Xa: num,Ys: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,cons_num(Xa,Ys))
    <=> ( ! [X5: num] :
            ( aa_set_num_o(member_num(X5),set_num2(Ys))
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X5) )
        & sorted_wrt_num(ord_less_eq_num,Ys) ) ) ).

% sorted_simps(2)
tff(fact_5199_sorted__simps_I2_J,axiom,
    ! [Xa: nat,Ys: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),cons_nat(Xa,Ys))
    <=> ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),aa_list_nat_set_nat(set_nat2,Ys))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X5) )
        & aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Ys) ) ) ).

% sorted_simps(2)
tff(fact_5200_sorted__simps_I2_J,axiom,
    ! [Xa: int,Ys: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,cons_int(Xa,Ys))
    <=> ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),aa_list_int_set_int(set_int2,Ys))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X5) )
        & sorted_wrt_int(ord_less_eq_int,Ys) ) ) ).

% sorted_simps(2)
tff(fact_5201_strict__sorted__simps_I2_J,axiom,
    ! [Xa: $o,Ys: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_o),cons_o((Xa),Ys))
    <=> ( ! [X5: $o] :
            ( aa_set_o_o(member_o((X5)),aa_list_o_set_o(set_o2,Ys))
           => aa_o_o(aa_o_fun_o_o(ord_less_o,(Xa)),(X5)) )
        & aa_list_o_o(sorted_wrt_o(ord_less_o),Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_5202_strict__sorted__simps_I2_J,axiom,
    ! [Xa: real,Ys: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_real),cons_real(Xa,Ys))
    <=> ( ! [X5: real] :
            ( aa_set_real_o(member_real(X5),aa_lis3769085826154824284t_real(set_real2,Ys))
           => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X5) )
        & aa_list_real_o(sorted_wrt_real(ord_less_real),Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_5203_strict__sorted__simps_I2_J,axiom,
    ! [Xa: rat,Ys: list_rat] :
      ( sorted_wrt_rat(ord_less_rat,cons_rat(Xa,Ys))
    <=> ( ! [X5: rat] :
            ( aa_set_rat_o(member_rat(X5),set_rat2(Ys))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X5) )
        & sorted_wrt_rat(ord_less_rat,Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_5204_strict__sorted__simps_I2_J,axiom,
    ! [Xa: num,Ys: list_num] :
      ( sorted_wrt_num(ord_less_num,cons_num(Xa,Ys))
    <=> ( ! [X5: num] :
            ( aa_set_num_o(member_num(X5),set_num2(Ys))
           => aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X5) )
        & sorted_wrt_num(ord_less_num,Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_5205_strict__sorted__simps_I2_J,axiom,
    ! [Xa: nat,Ys: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),cons_nat(Xa,Ys))
    <=> ( ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),aa_list_nat_set_nat(set_nat2,Ys))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X5) )
        & aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_5206_strict__sorted__simps_I2_J,axiom,
    ! [Xa: int,Ys: list_int] :
      ( sorted_wrt_int(ord_less_int,cons_int(Xa,Ys))
    <=> ( ! [X5: int] :
            ( aa_set_int_o(member_int(X5),aa_list_int_set_int(set_int2,Ys))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X5) )
        & sorted_wrt_int(ord_less_int,Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_5207_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_real),Xs)
     => aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_5208_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_rat,Xs)
     => sorted_wrt_rat(ord_less_eq_rat,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_5209_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_num,Xs)
     => sorted_wrt_num(ord_less_eq_num,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_5210_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Xs)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_5211_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int(ord_less_int,Xs)
     => sorted_wrt_int(ord_less_eq_int,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_5212_strict__sorted__equal,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_o),Xs)
     => ( aa_list_o_o(sorted_wrt_o(ord_less_o),Ys)
       => ( ( aa_list_o_set_o(set_o2,Ys) = aa_list_o_set_o(set_o2,Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_5213_strict__sorted__equal,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_real),Xs)
     => ( aa_list_real_o(sorted_wrt_real(ord_less_real),Ys)
       => ( ( aa_lis3769085826154824284t_real(set_real2,Ys) = aa_lis3769085826154824284t_real(set_real2,Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_5214_strict__sorted__equal,axiom,
    ! [Xs: list_rat,Ys: list_rat] :
      ( sorted_wrt_rat(ord_less_rat,Xs)
     => ( sorted_wrt_rat(ord_less_rat,Ys)
       => ( ( set_rat2(Ys) = set_rat2(Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_5215_strict__sorted__equal,axiom,
    ! [Xs: list_num,Ys: list_num] :
      ( sorted_wrt_num(ord_less_num,Xs)
     => ( sorted_wrt_num(ord_less_num,Ys)
       => ( ( set_num2(Ys) = set_num2(Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_5216_strict__sorted__equal,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Xs)
     => ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ys)
       => ( ( aa_list_nat_set_nat(set_nat2,Ys) = aa_list_nat_set_nat(set_nat2,Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_5217_strict__sorted__equal,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( sorted_wrt_int(ord_less_int,Xs)
     => ( sorted_wrt_int(ord_less_int,Ys)
       => ( ( aa_list_int_set_int(set_int2,Ys) = aa_list_int_set_int(set_int2,Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_5218_length__nth__simps_I2_J,axiom,
    ! [Xa: int,Xs: list_int] : ( size_size_list_int(cons_int(Xa,Xs)) = aa_nat_nat(suc,size_size_list_int(Xs)) ) ).

% length_nth_simps(2)
tff(fact_5219_length__nth__simps_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( size_s6755466524823107622T_VEBT(cons_VEBT_VEBT(Xa,Xs)) = aa_nat_nat(suc,size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_nth_simps(2)
tff(fact_5220_length__nth__simps_I2_J,axiom,
    ! [Xa: real,Xs: list_real] : ( size_size_list_real(cons_real(Xa,Xs)) = aa_nat_nat(suc,size_size_list_real(Xs)) ) ).

% length_nth_simps(2)
tff(fact_5221_length__nth__simps_I2_J,axiom,
    ! [Xa: $o,Xs: list_o] : ( size_size_list_o(cons_o((Xa),Xs)) = aa_nat_nat(suc,size_size_list_o(Xs)) ) ).

% length_nth_simps(2)
tff(fact_5222_length__nth__simps_I2_J,axiom,
    ! [Xa: nat,Xs: list_nat] : ( aa_list_nat_nat(size_size_list_nat,cons_nat(Xa,Xs)) = aa_nat_nat(suc,aa_list_nat_nat(size_size_list_nat,Xs)) ) ).

% length_nth_simps(2)
tff(fact_5223_length__Suc__conv,axiom,
    ! [Xs: list_int,Na: nat] :
      ( ( size_size_list_int(Xs) = aa_nat_nat(suc,Na) )
    <=> ? [Y5: int,Ys3: list_int] :
          ( ( Xs = cons_int(Y5,Ys3) )
          & ( size_size_list_int(Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_5224_length__Suc__conv,axiom,
    ! [Xs: list_VEBT_VEBT,Na: nat] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_nat_nat(suc,Na) )
    <=> ? [Y5: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(Y5,Ys3) )
          & ( size_s6755466524823107622T_VEBT(Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_5225_length__Suc__conv,axiom,
    ! [Xs: list_real,Na: nat] :
      ( ( size_size_list_real(Xs) = aa_nat_nat(suc,Na) )
    <=> ? [Y5: real,Ys3: list_real] :
          ( ( Xs = cons_real(Y5,Ys3) )
          & ( size_size_list_real(Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_5226_length__Suc__conv,axiom,
    ! [Xs: list_o,Na: nat] :
      ( ( size_size_list_o(Xs) = aa_nat_nat(suc,Na) )
    <=> ? [Y5: $o,Ys3: list_o] :
          ( ( Xs = cons_o((Y5),Ys3) )
          & ( size_size_list_o(Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_5227_length__Suc__conv,axiom,
    ! [Xs: list_nat,Na: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_nat_nat(suc,Na) )
    <=> ? [Y5: nat,Ys3: list_nat] :
          ( ( Xs = cons_nat(Y5,Ys3) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_5228_Suc__length__conv,axiom,
    ! [Na: nat,Xs: list_int] :
      ( ( aa_nat_nat(suc,Na) = size_size_list_int(Xs) )
    <=> ? [Y5: int,Ys3: list_int] :
          ( ( Xs = cons_int(Y5,Ys3) )
          & ( size_size_list_int(Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_5229_Suc__length__conv,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( ( aa_nat_nat(suc,Na) = size_s6755466524823107622T_VEBT(Xs) )
    <=> ? [Y5: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(Y5,Ys3) )
          & ( size_s6755466524823107622T_VEBT(Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_5230_Suc__length__conv,axiom,
    ! [Na: nat,Xs: list_real] :
      ( ( aa_nat_nat(suc,Na) = size_size_list_real(Xs) )
    <=> ? [Y5: real,Ys3: list_real] :
          ( ( Xs = cons_real(Y5,Ys3) )
          & ( size_size_list_real(Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_5231_Suc__length__conv,axiom,
    ! [Na: nat,Xs: list_o] :
      ( ( aa_nat_nat(suc,Na) = size_size_list_o(Xs) )
    <=> ? [Y5: $o,Ys3: list_o] :
          ( ( Xs = cons_o((Y5),Ys3) )
          & ( size_size_list_o(Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_5232_Suc__length__conv,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( ( aa_nat_nat(suc,Na) = aa_list_nat_nat(size_size_list_nat,Xs) )
    <=> ? [Y5: nat,Ys3: list_nat] :
          ( ( Xs = cons_nat(Y5,Ys3) )
          & ( aa_list_nat_nat(size_size_list_nat,Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_5233_impossible__Cons,axiom,
    ! [Xs: list_int,Ys: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),size_size_list_int(Ys))
     => ( Xs != cons_int(Xa,Ys) ) ) ).

% impossible_Cons
tff(fact_5234_impossible__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys))
     => ( Xs != cons_VEBT_VEBT(Xa,Ys) ) ) ).

% impossible_Cons
tff(fact_5235_impossible__Cons,axiom,
    ! [Xs: list_real,Ys: list_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),size_size_list_real(Ys))
     => ( Xs != cons_real(Xa,Ys) ) ) ).

% impossible_Cons
tff(fact_5236_impossible__Cons,axiom,
    ! [Xs: list_o,Ys: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),size_size_list_o(Ys))
     => ( Xs != cons_o((Xa),Ys) ) ) ).

% impossible_Cons
tff(fact_5237_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_list_nat_nat(size_size_list_nat,Ys))
     => ( Xs != cons_nat(Xa,Ys) ) ) ).

% impossible_Cons
tff(fact_5238_set__subset__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,cons_VEBT_VEBT(Xa,Xs))) ).

% set_subset_Cons
tff(fact_5239_set__subset__Cons,axiom,
    ! [Xs: list_real,Xa: real] : aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),aa_lis3769085826154824284t_real(set_real2,cons_real(Xa,Xs))) ).

% set_subset_Cons
tff(fact_5240_set__subset__Cons,axiom,
    ! [Xs: list_nat,Xa: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Xs)),aa_list_nat_set_nat(set_nat2,cons_nat(Xa,Xs))) ).

% set_subset_Cons
tff(fact_5241_set__subset__Cons,axiom,
    ! [Xs: list_o,Xa: $o] : aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Xs)),aa_list_o_set_o(set_o2,cons_o((Xa),Xs))) ).

% set_subset_Cons
tff(fact_5242_set__subset__Cons,axiom,
    ! [Xs: list_int,Xa: int] : aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Xs)),aa_list_int_set_int(set_int2,cons_int(Xa,Xs))) ).

% set_subset_Cons
tff(fact_5243_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A4: fun_VE2561185598974572172i_assn,A9: fun_VE2561185598974572172i_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                   => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                      & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi2),I2) )
                      & ( aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I2)) = aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A9,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L886525131989349516_VEBTi(I5,A4,Xs,Xsi) = vEBT_L886525131989349516_VEBTi(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5244_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_int,Xsi2: list_int,A4: fun_VE5641251438918192003t_assn,A9: fun_VE5641251438918192003t_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_int(Xsi) )
                   => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                      & ( aa_nat_int(nth_int(Xsi),I2) = aa_nat_int(nth_int(Xsi2),I2) )
                      & ( aa_int_assn(aa_VEB9043990991474615620t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)),aa_nat_int(nth_int(Xsi),I2)) = aa_int_assn(aa_VEB9043990991474615620t_assn(A9,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_int(nth_int(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L2806540629473551875Ti_int(I5,A4,Xs,Xsi) = vEBT_L2806540629473551875Ti_int(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5245_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A4: fun_in1903134756389476099i_assn,A9: fun_in1903134756389476099i_assn] :
      ( ( I5 = I6 )
     => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
       => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                 => ( ( size_size_list_int(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                   => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                      & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi2),I2) )
                      & ( aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(A4,aa_nat_int(nth_int(Xs),I2)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I2)) = aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(A9,aa_nat_int(nth_int(Xs4),I2)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L114188773329725699_VEBTi(I5,A4,Xs,Xsi) = vEBT_L114188773329725699_VEBTi(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5246_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_int,Xsi2: list_int,A4: fun_int_fun_int_assn,A9: fun_int_fun_int_assn] :
      ( ( I5 = I6 )
     => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
       => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                 => ( ( size_size_list_int(Xs) = size_size_list_int(Xsi) )
                   => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                      & ( aa_nat_int(nth_int(Xsi),I2) = aa_nat_int(nth_int(Xsi2),I2) )
                      & ( aa_int_assn(aa_int_fun_int_assn(A4,aa_nat_int(nth_int(Xs),I2)),aa_nat_int(nth_int(Xsi),I2)) = aa_int_assn(aa_int_fun_int_assn(A9,aa_nat_int(nth_int(Xs4),I2)),aa_nat_int(nth_int(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L8888932350013902202nt_int(I5,A4,Xs,Xsi) = vEBT_L8888932350013902202nt_int(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5247_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A4: fun_VE5577048379444288083T_assn,A9: fun_VE5577048379444288083T_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                      & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) )
                      & ( aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2)) = aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A9,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L2497118539674116125T_VEBT(I5,A4,Xs,Xsi) = vEBT_L2497118539674116125T_VEBT(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5248_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A4: fun_in7755481285845205148T_assn,A9: fun_in7755481285845205148T_assn] :
      ( ( I5 = I6 )
     => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                 => ( ( size_size_list_int(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                      & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) )
                      & ( aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A4,aa_nat_int(nth_int(Xs),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2)) = aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A9,aa_nat_int(nth_int(Xs4),I2)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L2018189785592951398T_VEBT(I5,A4,Xs,Xsi) = vEBT_L2018189785592951398T_VEBT(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5249_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real,A4: fun_VE1885522831824361987l_assn,A9: fun_VE1885522831824361987l_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_real(Xsi) )
                   => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                      & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) )
                      & ( aa_real_assn(aa_VEB9107798209145535940l_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)),aa_nat_real(nth_real(Xsi),I2)) = aa_real_assn(aa_VEB9107798209145535940l_assn(A9,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_real(nth_real(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L7728200936804140803i_real(I5,A4,Xs,Xsi) = vEBT_L7728200936804140803i_real(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5250_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_real,Xsi2: list_real,A4: fun_in6446958447386763770l_assn,A9: fun_in6446958447386763770l_assn] :
      ( ( I5 = I6 )
     => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
       => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                 => ( ( size_size_list_int(Xs) = size_size_list_real(Xsi) )
                   => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                      & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) )
                      & ( aa_real_assn(aa_int_fun_real_assn(A4,aa_nat_int(nth_int(Xs),I2)),aa_nat_real(nth_real(Xsi),I2)) = aa_real_assn(aa_int_fun_real_assn(A9,aa_nat_int(nth_int(Xs4),I2)),aa_nat_real(nth_real(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L7077748017936769786t_real(I5,A4,Xs,Xsi) = vEBT_L7077748017936769786t_real(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5251_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_o,Xsi2: list_o,A4: fun_VE9201042017749490615o_assn,A9: fun_VE9201042017749490615o_assn] :
      ( ( I5 = I6 )
     => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
       => ( ( size_size_list_o(Xsi) = size_size_list_o(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                 => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_o(Xsi) )
                   => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                      & ( aa_nat_o(nth_o(Xsi),I2)
                      <=> aa_nat_o(nth_o(Xsi2),I2) )
                      & ( aa_o_assn(aa_VEB6687300313434771872o_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2)),aa_nat_o(nth_o(Xsi),I2)) = aa_o_assn(aa_VEB6687300313434771872o_assn(A9,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2)),aa_nat_o(nth_o(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L3328983362619735041EBTi_o(I5,A4,Xs,Xsi) = vEBT_L3328983362619735041EBTi_o(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5252_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_int,Xs4: list_int,Xsi: list_o,Xsi2: list_o,A4: fun_int_fun_o_assn,A9: fun_int_fun_o_assn] :
      ( ( I5 = I6 )
     => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
       => ( ( size_size_list_o(Xsi) = size_size_list_o(Xsi2) )
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                 => ( ( size_size_list_int(Xs) = size_size_list_o(Xsi) )
                   => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                      & ( aa_nat_o(nth_o(Xsi),I2)
                      <=> aa_nat_o(nth_o(Xsi2),I2) )
                      & ( aa_o_assn(aa_int_fun_o_assn(A4,aa_nat_int(nth_int(Xs),I2)),aa_nat_o(nth_o(Xsi),I2)) = aa_o_assn(aa_int_fun_o_assn(A9,aa_nat_int(nth_int(Xs4),I2)),aa_nat_o(nth_o(Xsi2),I2)) ) ) ) ) )
           => ( vEBT_L3563379889750563018_int_o(I5,A4,Xs,Xsi) = vEBT_L3563379889750563018_int_o(I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5253_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_VE2561185598974572172i_assn,A9: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                     => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                        & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L886525131989349516_VEBTi(I5,A4,Xs,Xsi) = vEBT_L886525131989349516_VEBTi(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5254_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_VE5641251438918192003t_assn,A9: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_int,Xsi2: list_int] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_int(Xsi) )
                     => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                        & ( aa_nat_int(nth_int(Xsi),I2) = aa_nat_int(nth_int(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L2806540629473551875Ti_int(I5,A4,Xs,Xsi) = vEBT_L2806540629473551875Ti_int(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5255_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_in1903134756389476099i_assn,A9: fun_in1903134756389476099i_assn,Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
         => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                   => ( ( size_size_list_int(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                     => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                        & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L114188773329725699_VEBTi(I5,A4,Xs,Xsi) = vEBT_L114188773329725699_VEBTi(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5256_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_int_fun_int_assn,A9: fun_int_fun_int_assn,Xs: list_int,Xs4: list_int,Xsi: list_int,Xsi2: list_int] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
         => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                   => ( ( size_size_list_int(Xs) = size_size_list_int(Xsi) )
                     => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                        & ( aa_nat_int(nth_int(Xsi),I2) = aa_nat_int(nth_int(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L8888932350013902202nt_int(I5,A4,Xs,Xsi) = vEBT_L8888932350013902202nt_int(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5257_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_VE5577048379444288083T_assn,A9: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                        & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L2497118539674116125T_VEBT(I5,A4,Xs,Xsi) = vEBT_L2497118539674116125T_VEBT(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5258_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_in7755481285845205148T_assn,A9: fun_in7755481285845205148T_assn,Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                   => ( ( size_size_list_int(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                        & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L2018189785592951398T_VEBT(I5,A4,Xs,Xsi) = vEBT_L2018189785592951398T_VEBT(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5259_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_VE1885522831824361987l_assn,A9: fun_VE1885522831824361987l_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_real(Xsi) )
                     => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                        & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L7728200936804140803i_real(I5,A4,Xs,Xsi) = vEBT_L7728200936804140803i_real(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5260_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_in6446958447386763770l_assn,A9: fun_in6446958447386763770l_assn,Xs: list_int,Xs4: list_int,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
         => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                   => ( ( size_size_list_int(Xs) = size_size_list_real(Xsi) )
                     => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                        & ( aa_nat_real(nth_real(Xsi),I2) = aa_nat_real(nth_real(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L7077748017936769786t_real(I5,A4,Xs,Xsi) = vEBT_L7077748017936769786t_real(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5261_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_VE9201042017749490615o_assn,A9: fun_VE9201042017749490615o_assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_o,Xsi2: list_o] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Xs4) )
         => ( ( size_size_list_o(Xsi) = size_size_list_o(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
                   => ( ( size_s7982070591426661849_VEBTi(Xs) = size_size_list_o(Xsi) )
                     => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs4),I2) )
                        & ( aa_nat_o(nth_o(Xsi),I2)
                        <=> aa_nat_o(nth_o(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L3328983362619735041EBTi_o(I5,A4,Xs,Xsi) = vEBT_L3328983362619735041EBTi_o(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5262_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A4: fun_int_fun_o_assn,A9: fun_int_fun_o_assn,Xs: list_int,Xs4: list_int,Xsi: list_o,Xsi2: list_o] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( size_size_list_int(Xs) = size_size_list_int(Xs4) )
         => ( ( size_size_list_o(Xsi) = size_size_list_o(Xsi2) )
           => ( ! [I2: nat] :
                  ( aa_set_nat_o(member_nat(I2),I5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
                   => ( ( size_size_list_int(Xs) = size_size_list_o(Xsi) )
                     => ( ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Xs4),I2) )
                        & ( aa_nat_o(nth_o(Xsi),I2)
                        <=> aa_nat_o(nth_o(Xsi2),I2) ) ) ) ) )
             => ( vEBT_L3563379889750563018_int_o(I5,A4,Xs,Xsi) = vEBT_L3563379889750563018_int_o(I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5263_sorted__wrt01,axiom,
    ! [Xs: list_int,P2: fun_int_fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => sorted_wrt_int(P2,Xs) ) ).

% sorted_wrt01
tff(fact_5264_sorted__wrt01,axiom,
    ! [Xs: list_VEBT_VEBT,P2: fun_VE2962686792283380846VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat)
     => aa_list_VEBT_VEBT_o(sorted_wrt_VEBT_VEBT(P2),Xs) ) ).

% sorted_wrt01
tff(fact_5265_sorted__wrt01,axiom,
    ! [Xs: list_real,P2: fun_real_fun_real_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),one_one_nat)
     => aa_list_real_o(sorted_wrt_real(P2),Xs) ) ).

% sorted_wrt01
tff(fact_5266_sorted__wrt01,axiom,
    ! [Xs: list_o,P2: fun_o_fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => aa_list_o_o(sorted_wrt_o(P2),Xs) ) ).

% sorted_wrt01
tff(fact_5267_sorted__wrt01,axiom,
    ! [Xs: list_nat,P2: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => aa_list_nat_o(sorted_wrt_nat(P2),Xs) ) ).

% sorted_wrt01
tff(fact_5268_sorted__wrt__iff__nth__less,axiom,
    ! [P2: fun_VE8636387394336989792EBTi_o,Xs: list_VEBT_VEBTi] :
      ( sorted9206477368072086664_VEBTi(P2,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_s7982070591426661849_VEBTi(Xs))
           => aa_VEBT_VEBTi_o(aa_VEB8157431426570739913EBTi_o(P2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I3)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5269_sorted__wrt__iff__nth__less,axiom,
    ! [P2: fun_int_fun_int_o,Xs: list_int] :
      ( sorted_wrt_int(P2,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(P2,aa_nat_int(nth_int(Xs),I3)),aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5270_sorted__wrt__iff__nth__less,axiom,
    ! [P2: fun_VE2962686792283380846VEBT_o,Xs: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(sorted_wrt_VEBT_VEBT(P2),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_s6755466524823107622T_VEBT(Xs))
           => aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(P2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5271_sorted__wrt__iff__nth__less,axiom,
    ! [P2: fun_real_fun_real_o,Xs: list_real] :
      ( aa_list_real_o(sorted_wrt_real(P2),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_real(Xs))
           => aa_real_o(aa_real_fun_real_o(P2,aa_nat_real(nth_real(Xs),I3)),aa_nat_real(nth_real(Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5272_sorted__wrt__iff__nth__less,axiom,
    ! [P2: fun_o_fun_o_o,Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(P2),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_o(Xs))
           => aa_o_o(aa_o_fun_o_o(P2,aa_nat_o(nth_o(Xs),I3)),aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5273_sorted__wrt__iff__nth__less,axiom,
    ! [P2: fun_nat_fun_nat_o,Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(P2),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(P2,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_5274_sorted__wrt__nth__less,axiom,
    ! [P2: fun_VE8636387394336989792EBTi_o,Xs: list_VEBT_VEBTi,I: nat,J: nat] :
      ( sorted9206477368072086664_VEBTi(P2,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(aa_VEB8157431426570739913EBTi_o(P2,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5275_sorted__wrt__nth__less,axiom,
    ! [P2: fun_int_fun_int_o,Xs: list_int,I: nat,J: nat] :
      ( sorted_wrt_int(P2,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(P2,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5276_sorted__wrt__nth__less,axiom,
    ! [P2: fun_VE2962686792283380846VEBT_o,Xs: list_VEBT_VEBT,I: nat,J: nat] :
      ( aa_list_VEBT_VEBT_o(sorted_wrt_VEBT_VEBT(P2),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(P2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5277_sorted__wrt__nth__less,axiom,
    ! [P2: fun_real_fun_real_o,Xs: list_real,I: nat,J: nat] :
      ( aa_list_real_o(sorted_wrt_real(P2),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(Xs))
         => aa_real_o(aa_real_fun_real_o(P2,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5278_sorted__wrt__nth__less,axiom,
    ! [P2: fun_o_fun_o_o,Xs: list_o,I: nat,J: nat] :
      ( aa_list_o_o(sorted_wrt_o(P2),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
         => aa_o_o(aa_o_fun_o_o(P2,aa_nat_o(nth_o(Xs),I)),aa_nat_o(nth_o(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5279_sorted__wrt__nth__less,axiom,
    ! [P2: fun_nat_fun_nat_o,Xs: list_nat,I: nat,J: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(P2),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(P2,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_5280_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_5281_Suc__le__length__iff,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),size_size_list_int(Xs))
    <=> ? [X5: int,Ys3: list_int] :
          ( ( Xs = cons_int(X5,Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_int(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_5282_Suc__le__length__iff,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),size_s6755466524823107622T_VEBT(Xs))
    <=> ? [X5: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(X5,Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s6755466524823107622T_VEBT(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_5283_Suc__le__length__iff,axiom,
    ! [Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),size_size_list_real(Xs))
    <=> ? [X5: real,Ys3: list_real] :
          ( ( Xs = cons_real(X5,Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_real(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_5284_Suc__le__length__iff,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),size_size_list_o(Xs))
    <=> ? [X5: $o,Ys3: list_o] :
          ( ( Xs = cons_o((X5),Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_o(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_5285_Suc__le__length__iff,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_list_nat_nat(size_size_list_nat,Xs))
    <=> ? [X5: nat,Ys3: list_nat] :
          ( ( Xs = cons_nat(X5,Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_list_nat_nat(size_size_list_nat,Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_5286_sorted01,axiom,
    ! [Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),one_one_nat)
     => aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Xs) ) ).

% sorted01
tff(fact_5287_sorted01,axiom,
    ! [Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs) ) ).

% sorted01
tff(fact_5288_sorted01,axiom,
    ! [Xs: list_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_rat(Xs)),one_one_nat)
     => sorted_wrt_rat(ord_less_eq_rat,Xs) ) ).

% sorted01
tff(fact_5289_sorted01,axiom,
    ! [Xs: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_num(Xs)),one_one_nat)
     => sorted_wrt_num(ord_less_eq_num,Xs) ) ).

% sorted01
tff(fact_5290_sorted01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),one_one_nat)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs) ) ).

% sorted01
tff(fact_5291_sorted01,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => sorted_wrt_int(ord_less_eq_int,Xs) ) ).

% sorted01
tff(fact_5292_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_real(Xs))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(nth_real(Xs),I3)),aa_nat_real(nth_real(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5293_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_o(Xs))
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I3)),aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5294_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_rat(Xs))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I3)),aa_nat_rat(nth_rat(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5295_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_num(Xs))
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I3)),aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5296_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5297_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I3)),aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_5298_list_Osize_I4_J,axiom,
    ! [X21: int,X22: list_int] : ( size_size_list_int(cons_int(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(X22)),aa_nat_nat(suc,zero_zero_nat)) ) ).

% list.size(4)
tff(fact_5299_list_Osize_I4_J,axiom,
    ! [X21: vEBT_VEBT,X22: list_VEBT_VEBT] : ( size_s6755466524823107622T_VEBT(cons_VEBT_VEBT(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6755466524823107622T_VEBT(X22)),aa_nat_nat(suc,zero_zero_nat)) ) ).

% list.size(4)
tff(fact_5300_list_Osize_I4_J,axiom,
    ! [X21: real,X22: list_real] : ( size_size_list_real(cons_real(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_real(X22)),aa_nat_nat(suc,zero_zero_nat)) ) ).

% list.size(4)
tff(fact_5301_list_Osize_I4_J,axiom,
    ! [X21: $o,X22: list_o] : ( size_size_list_o(cons_o((X21),X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_o(X22)),aa_nat_nat(suc,zero_zero_nat)) ) ).

% list.size(4)
tff(fact_5302_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] : ( aa_list_nat_nat(size_size_list_nat,cons_nat(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(size_size_list_nat,X22)),aa_nat_nat(suc,zero_zero_nat)) ) ).

% list.size(4)
tff(fact_5303_nth__Cons_H,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat] :
      ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Xa,Xs)),Na) = $ite(Na = zero_zero_nat,Xa,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% nth_Cons'
tff(fact_5304_nth__Cons_H,axiom,
    ! [Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi,Na: nat] :
      ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Xa,Xs)),Na) = $ite(Na = zero_zero_nat,Xa,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% nth_Cons'
tff(fact_5305_nth__Cons_H,axiom,
    ! [Xa: int,Xs: list_int,Na: nat] :
      ( aa_nat_int(nth_int(cons_int(Xa,Xs)),Na) = $ite(Na = zero_zero_nat,Xa,aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% nth_Cons'
tff(fact_5306_nth__Cons_H,axiom,
    ! [Xa: nat,Xs: list_nat,Na: nat] :
      ( aa_nat_nat(nth_nat(cons_nat(Xa,Xs)),Na) = $ite(Na = zero_zero_nat,Xa,aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% nth_Cons'
tff(fact_5307_nth__Cons_H,axiom,
    ! [Xa: $o,Xs: list_o,Na: nat] :
      ( aa_nat_o(nth_o(cons_o((Xa),Xs)),Na)
    <=> $ite(Na = zero_zero_nat,(Xa),aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ).

% nth_Cons'
tff(fact_5308_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE5577048379444288083T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5309_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE1225457145571040039t_assn,Xsi: list_nat] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2809031099982602151Ti_nat(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5310_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE2561185598974572172i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5311_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE5641251438918192003t_assn,Xsi: list_int] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2806540629473551875Ti_int(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5312_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: fun_in7755481285845205148T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( vEBT_L2018189785592951398T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2018189785592951398T_VEBT(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5313_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: fun_int_fun_nat_assn,Xsi: list_nat] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( vEBT_L8891422820522952478nt_nat(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8891422820522952478nt_nat(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5314_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: fun_in1903134756389476099i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( vEBT_L114188773329725699_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L114188773329725699_VEBTi(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5315_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A4: fun_int_fun_int_assn,Xsi: list_int] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( vEBT_L8888932350013902202nt_int(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8888932350013902202nt_int(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5316_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5317_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L8650695023172932196BT_nat(aa_set_nat_set_nat(insert_nat(I),I5),A4,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5318_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I3)),size_size_list_real(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(nth_real(Xs),I3)),aa_nat_real(nth_real(Xs),aa_nat_nat(suc,I3))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5319_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I3)),size_size_list_o(Xs))
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I3)),aa_nat_o(nth_o(Xs),aa_nat_nat(suc,I3))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5320_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I3)),size_size_list_rat(Xs))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I3)),aa_nat_rat(nth_rat(Xs),aa_nat_nat(suc,I3))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5321_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I3)),size_size_list_num(Xs))
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I3)),aa_nat_num(nth_num(Xs),aa_nat_nat(suc,I3))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5322_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I3)),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,I3))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5323_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I3)),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I3)),aa_nat_int(nth_int(Xs),aa_nat_nat(suc,I3))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_5324_sorted__iff__nth__mono,axiom,
    ! [Xs: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_real(Xs))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(nth_real(Xs),I3)),aa_nat_real(nth_real(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5325_sorted__iff__nth__mono,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_o(Xs))
           => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I3)),aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5326_sorted__iff__nth__mono,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_rat(Xs))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I3)),aa_nat_rat(nth_rat(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5327_sorted__iff__nth__mono,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_num(Xs))
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I3)),aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5328_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I3)),aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5329_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_int(Xs))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I3)),aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_5330_sorted__nth__mono,axiom,
    ! [Xs: list_real,I: nat,J: nat] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(nth_real(Xs),I)),aa_nat_real(nth_real(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_5331_sorted__nth__mono,axiom,
    ! [Xs: list_o,I: nat,J: nat] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
         => aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(Xs),I)),aa_nat_o(nth_o(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_5332_sorted__nth__mono,axiom,
    ! [Xs: list_rat,I: nat,J: nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_rat(Xs))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(Xs),I)),aa_nat_rat(nth_rat(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_5333_sorted__nth__mono,axiom,
    ! [Xs: list_num,I: nat,J: nat] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_num(Xs))
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(Xs),I)),aa_nat_num(nth_num(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_5334_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_5335_sorted__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J)) ) ) ) ).

% sorted_nth_mono
tff(fact_5336_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_literal] :
      ( finite5847741373460823677iteral(A4)
     => ~ ! [L2: list_literal] :
            ( sorted_wrt_literal(ord_less_literal,L2)
           => ( ( set_literal2(L2) = A4 )
             => ( size_s2501651207091587910iteral(L2) != finite_card_literal(A4) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5337_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ~ ! [L2: list_Code_integer] :
            ( sorted710888440204495920nteger(ord_le6747313008572928689nteger,L2)
           => ( ( set_Code_integer2(L2) = A4 )
             => ( size_s3445333598471063425nteger(L2) != finite4902975817058060853nteger(A4) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5338_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ~ ! [L2: list_o] :
            ( aa_list_o_o(sorted_wrt_o(ord_less_o),L2)
           => ( ( aa_list_o_set_o(set_o2,L2) = A4 )
             => ( size_size_list_o(L2) != finite_card_o(A4) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5339_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ~ ! [L2: list_real] :
            ( aa_list_real_o(sorted_wrt_real(ord_less_real),L2)
           => ( ( aa_lis3769085826154824284t_real(set_real2,L2) = A4 )
             => ( size_size_list_real(L2) != finite_card_real(A4) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5340_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ~ ! [L2: list_rat] :
            ( sorted_wrt_rat(ord_less_rat,L2)
           => ( ( set_rat2(L2) = A4 )
             => ( size_size_list_rat(L2) != finite_card_rat(A4) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5341_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ~ ! [L2: list_num] :
            ( sorted_wrt_num(ord_less_num,L2)
           => ( ( set_num2(L2) = A4 )
             => ( size_size_list_num(L2) != finite_card_num(A4) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5342_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ~ ! [L2: list_nat] :
            ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),L2)
           => ( ( aa_list_nat_set_nat(set_nat2,L2) = A4 )
             => ( aa_list_nat_nat(size_size_list_nat,L2) != aa_set_nat_nat(finite_card_nat,A4) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5343_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ~ ! [L2: list_int] :
            ( sorted_wrt_int(ord_less_int,L2)
           => ( ( aa_list_int_set_int(set_int2,L2) = A4 )
             => ( size_size_list_int(L2) != finite_card_int(A4) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_5344_nth__non__equal__first__eq,axiom,
    ! [Xa: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat] :
      ( ( Xa != Y )
     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Xa,Xs)),Na) = Y )
      <=> ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5345_nth__non__equal__first__eq,axiom,
    ! [Xa: vEBT_VEBTi,Y: vEBT_VEBTi,Xs: list_VEBT_VEBTi,Na: nat] :
      ( ( Xa != Y )
     => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Xa,Xs)),Na) = Y )
      <=> ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5346_nth__non__equal__first__eq,axiom,
    ! [Xa: int,Y: int,Xs: list_int,Na: nat] :
      ( ( Xa != Y )
     => ( ( aa_nat_int(nth_int(cons_int(Xa,Xs)),Na) = Y )
      <=> ( ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5347_nth__non__equal__first__eq,axiom,
    ! [Xa: nat,Y: nat,Xs: list_nat,Na: nat] :
      ( ( Xa != Y )
     => ( ( aa_nat_nat(nth_nat(cons_nat(Xa,Xs)),Na) = Y )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5348_nth__non__equal__first__eq,axiom,
    ! [Xa: $o,Y: $o,Xs: list_o,Na: nat] :
      ( ~ ( (Xa)
        <=> (Y) )
     => ( ( aa_nat_o(nth_o(cons_o((Xa),Xs)),Na)
        <=> (Y) )
      <=> ( ( aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))
          <=> (Y) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5349_nth__equal__first__eq,axiom,
    ! [Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi,Na: nat] :
      ( ~ member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s7982070591426661849_VEBTi(Xs))
       => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(cons_VEBT_VEBTi(Xa,Xs)),Na) = Xa )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5350_nth__equal__first__eq,axiom,
    ! [Xa: set_nat,Xs: list_set_nat,Na: nat] :
      ( ~ aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s3254054031482475050et_nat(Xs))
       => ( ( aa_nat_set_nat(nth_set_nat(cons_set_nat(Xa,Xs)),Na) = Xa )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5351_nth__equal__first__eq,axiom,
    ! [Xa: int,Xs: list_int,Na: nat] :
      ( ~ aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_int(Xs))
       => ( ( aa_nat_int(nth_int(cons_int(Xa,Xs)),Na) = Xa )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5352_nth__equal__first__eq,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s6755466524823107622T_VEBT(Xs))
       => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(cons_VEBT_VEBT(Xa,Xs)),Na) = Xa )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5353_nth__equal__first__eq,axiom,
    ! [Xa: real,Xs: list_real,Na: nat] :
      ( ~ aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_real(Xs))
       => ( ( aa_nat_real(nth_real(cons_real(Xa,Xs)),Na) = Xa )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5354_nth__equal__first__eq,axiom,
    ! [Xa: $o,Xs: list_o,Na: nat] :
      ( ~ aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_o(Xs))
       => ( ( aa_nat_o(nth_o(cons_o((Xa),Xs)),Na)
          <=> (Xa) )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5355_nth__equal__first__eq,axiom,
    ! [Xa: nat,Xs: list_nat,Na: nat] :
      ( ~ aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( ( aa_nat_nat(nth_nat(cons_nat(Xa,Xs)),Na) = Xa )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5356_slice__Cons,axiom,
    ! [Begin: nat,End: nat,Xa: int,Xs: list_int] :
      ( slice_int(Begin,End,cons_int(Xa,Xs)) = $ite(
          ( ( Begin = zero_zero_nat )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
          cons_int(Xa,slice_int(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
          slice_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ) ).

% slice_Cons
tff(fact_5357_slice__Cons,axiom,
    ! [Begin: nat,End: nat,Xa: nat,Xs: list_nat] :
      ( slice_nat(Begin,End,cons_nat(Xa,Xs)) = $ite(
          ( ( Begin = zero_zero_nat )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
          cons_nat(Xa,slice_nat(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
          slice_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ) ).

% slice_Cons
tff(fact_5358_slice__Cons,axiom,
    ! [Begin: nat,End: nat,Xa: $o,Xs: list_o] :
      ( slice_o(Begin,End,cons_o((Xa),Xs)) = $ite(
          ( ( Begin = zero_zero_nat )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
          cons_o((Xa),slice_o(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
          slice_o(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ) ).

% slice_Cons
tff(fact_5359_length__Cons,axiom,
    ! [Xa: int,Xs: list_int] : ( size_size_list_int(cons_int(Xa,Xs)) = aa_nat_nat(suc,size_size_list_int(Xs)) ) ).

% length_Cons
tff(fact_5360_length__Cons,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( size_s6755466524823107622T_VEBT(cons_VEBT_VEBT(Xa,Xs)) = aa_nat_nat(suc,size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_Cons
tff(fact_5361_length__Cons,axiom,
    ! [Xa: real,Xs: list_real] : ( size_size_list_real(cons_real(Xa,Xs)) = aa_nat_nat(suc,size_size_list_real(Xs)) ) ).

% length_Cons
tff(fact_5362_length__Cons,axiom,
    ! [Xa: $o,Xs: list_o] : ( size_size_list_o(cons_o((Xa),Xs)) = aa_nat_nat(suc,size_size_list_o(Xs)) ) ).

% length_Cons
tff(fact_5363_length__Cons,axiom,
    ! [Xa: nat,Xs: list_nat] : ( aa_list_nat_nat(size_size_list_nat,cons_nat(Xa,Xs)) = aa_nat_nat(suc,aa_list_nat_nat(size_size_list_nat,Xs)) ) ).

% length_Cons
tff(fact_5364_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2497118539674116125T_VEBT(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5365_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2809031099982602151Ti_nat(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5366_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L886525131989349516_VEBTi(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5367_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2806540629473551875Ti_int(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5368_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_in7755481285845205148T_assn,Xs: list_int,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2018189785592951398T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2018189785592951398T_VEBT(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5369_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_int_fun_nat_assn,Xs: list_int,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8891422820522952478nt_nat(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8891422820522952478nt_nat(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5370_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_in1903134756389476099i_assn,Xs: list_int,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L114188773329725699_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L114188773329725699_VEBTi(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5371_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_int_fun_int_assn,Xs: list_int,I: nat,Xsi: list_int,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8888932350013902202nt_int(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8888932350013902202nt_int(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5372_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3204528365124325536T_VEBT(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5373_listI__assn__reinsert,axiom,
    ! [P2: assn,A4: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8650695023172932196BT_nat(I5,A4,Xs,Xsi)),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5374_sorted__list__of__set__nonempty,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( linord2324613341767563021nteger(A4) = cons_Code_integer(lattic1063845414844153500nteger(A4),linord2324613341767563021nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(lattic1063845414844153500nteger(A4),bot_bo3990330152332043303nteger)))) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_5375_sorted__list__of__set__nonempty,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( linord2612477271533052124et_int(A4) = cons_int(lattic8718645017227715691in_int(A4),linord2612477271533052124et_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(lattic8718645017227715691in_int(A4)),bot_bot_set_int)))) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_5376_sorted__list__of__set__nonempty,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( linord3142498349692569832_set_o(A4) = cons_o(aa_set_o_o(lattic1973801136483472281_Min_o,A4),linord3142498349692569832_set_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o(aa_set_o_o(lattic1973801136483472281_Min_o,A4)),bot_bot_set_o)))) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_5377_sorted__list__of__set__nonempty,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( linord2614967742042102400et_nat(A4) = cons_nat(aa_set_nat_nat(lattic8721135487736765967in_nat,A4),linord2614967742042102400et_nat(minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(aa_set_nat_nat(lattic8721135487736765967in_nat,A4)),bot_bot_set_nat)))) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_5378_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_literal,L: list_literal] :
      ( finite5847741373460823677iteral(A4)
     => ( ( sorted_wrt_literal(ord_less_literal,L)
          & ( set_literal2(L) = A4 )
          & ( size_s2501651207091587910iteral(L) = finite_card_literal(A4) ) )
      <=> ( linord2913955441264437540iteral(A4) = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5379_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_Code_integer,L: list_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( sorted710888440204495920nteger(ord_le6747313008572928689nteger,L)
          & ( set_Code_integer2(L) = A4 )
          & ( size_s3445333598471063425nteger(L) = finite4902975817058060853nteger(A4) ) )
      <=> ( linord2324613341767563021nteger(A4) = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5380_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_o,L: list_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( aa_list_o_o(sorted_wrt_o(ord_less_o),L)
          & ( aa_list_o_set_o(set_o2,L) = A4 )
          & ( size_size_list_o(L) = finite_card_o(A4) ) )
      <=> ( linord3142498349692569832_set_o(A4) = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5381_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_real,L: list_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( ( aa_list_real_o(sorted_wrt_real(ord_less_real),L)
          & ( aa_lis3769085826154824284t_real(set_real2,L) = A4 )
          & ( size_size_list_real(L) = finite_card_real(A4) ) )
      <=> ( linord4252657396651189596t_real(A4) = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5382_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_rat,L: list_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ( ( sorted_wrt_rat(ord_less_rat,L)
          & ( set_rat2(L) = A4 )
          & ( size_size_list_rat(L) = finite_card_rat(A4) ) )
      <=> ( linord1979837681955606664et_rat(A4) = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5383_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_num,L: list_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ( ( sorted_wrt_num(ord_less_num,L)
          & ( set_num2(L) = A4 )
          & ( size_size_list_num(L) = finite_card_num(A4) ) )
      <=> ( linord8395671565052656842et_num(A4) = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5384_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_int,L: list_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( sorted_wrt_int(ord_less_int,L)
          & ( aa_list_int_set_int(set_int2,L) = A4 )
          & ( size_size_list_int(L) = finite_card_int(A4) ) )
      <=> ( linord2612477271533052124et_int(A4) = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5385_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A4: set_nat,L: list_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),L)
          & ( aa_list_nat_set_nat(set_nat2,L) = A4 )
          & ( aa_list_nat_nat(size_size_list_nat,L) = aa_set_nat_nat(finite_card_nat,A4) ) )
      <=> ( linord2614967742042102400et_nat(A4) = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_5386_converse__trancl__induct,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32,P2: fun_uint32_o] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),transi3114468042090999947uint32(R))
     => ( ! [Y3: uint32] :
            ( member8027108493173000802uint32(produc1400373151660368625uint32(Y3,B),R)
           => aa_uint32_o(P2,Y3) )
       => ( ! [Y3: uint32,Z3: uint32] :
              ( member8027108493173000802uint32(produc1400373151660368625uint32(Y3,Z3),R)
             => ( member8027108493173000802uint32(produc1400373151660368625uint32(Z3,B),transi3114468042090999947uint32(R))
               => ( aa_uint32_o(P2,Z3)
                 => aa_uint32_o(P2,Y3) ) ) )
         => aa_uint32_o(P2,A) ) ) ) ).

% converse_trancl_induct
tff(fact_5387_converse__trancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P2: fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),transi6264000038957366511cl_nat(R))
     => ( ! [Y3: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),B),R)
           => aa_nat_o(P2,Y3) )
       => ( ! [Y3: nat,Z3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z3),R)
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Z3),B),transi6264000038957366511cl_nat(R))
               => ( aa_nat_o(P2,Z3)
                 => aa_nat_o(P2,Y3) ) ) )
         => aa_nat_o(P2,A) ) ) ) ).

% converse_trancl_induct
tff(fact_5388_converse__trancl__induct,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,P2: fun_int_o] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),transi6261509568448316235cl_int(R))
     => ( ! [Y3: int] :
            ( member5262025264175285858nt_int(product_Pair_int_int(Y3,B),R)
           => aa_int_o(P2,Y3) )
       => ( ! [Y3: int,Z3: int] :
              ( member5262025264175285858nt_int(product_Pair_int_int(Y3,Z3),R)
             => ( member5262025264175285858nt_int(product_Pair_int_int(Z3,B),transi6261509568448316235cl_int(R))
               => ( aa_int_o(P2,Z3)
                 => aa_int_o(P2,Y3) ) ) )
         => aa_int_o(P2,A) ) ) ) ).

% converse_trancl_induct
tff(fact_5389_trancl__trans__induct,axiom,
    ! [Xa: uint32,Y: uint32,R: set_Pr1773385645901665561uint32,P2: fun_ui3478071520421546738nt32_o] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y),transi3114468042090999947uint32(R))
     => ( ! [X: uint32,Y3: uint32] :
            ( member8027108493173000802uint32(produc1400373151660368625uint32(X,Y3),R)
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,X),Y3) )
       => ( ! [X: uint32,Y3: uint32,Z3: uint32] :
              ( member8027108493173000802uint32(produc1400373151660368625uint32(X,Y3),transi3114468042090999947uint32(R))
             => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,X),Y3)
               => ( member8027108493173000802uint32(produc1400373151660368625uint32(Y3,Z3),transi3114468042090999947uint32(R))
                 => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,Y3),Z3)
                   => aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,X),Z3) ) ) ) )
         => aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,Xa),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_5390_trancl__trans__induct,axiom,
    ! [Xa: nat,Y: nat,R: set_Pr1261947904930325089at_nat,P2: fun_nat_fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y),transi6264000038957366511cl_nat(R))
     => ( ! [X: nat,Y3: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3),R)
           => aa_nat_o(aa_nat_fun_nat_o(P2,X),Y3) )
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3),transi6264000038957366511cl_nat(R))
             => ( aa_nat_o(aa_nat_fun_nat_o(P2,X),Y3)
               => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z3),transi6264000038957366511cl_nat(R))
                 => ( aa_nat_o(aa_nat_fun_nat_o(P2,Y3),Z3)
                   => aa_nat_o(aa_nat_fun_nat_o(P2,X),Z3) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,Xa),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_5391_trancl__trans__induct,axiom,
    ! [Xa: int,Y: int,R: set_Pr958786334691620121nt_int,P2: fun_int_fun_int_o] :
      ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Y),transi6261509568448316235cl_int(R))
     => ( ! [X: int,Y3: int] :
            ( member5262025264175285858nt_int(product_Pair_int_int(X,Y3),R)
           => aa_int_o(aa_int_fun_int_o(P2,X),Y3) )
       => ( ! [X: int,Y3: int,Z3: int] :
              ( member5262025264175285858nt_int(product_Pair_int_int(X,Y3),transi6261509568448316235cl_int(R))
             => ( aa_int_o(aa_int_fun_int_o(P2,X),Y3)
               => ( member5262025264175285858nt_int(product_Pair_int_int(Y3,Z3),transi6261509568448316235cl_int(R))
                 => ( aa_int_o(aa_int_fun_int_o(P2,Y3),Z3)
                   => aa_int_o(aa_int_fun_int_o(P2,X),Z3) ) ) ) )
         => aa_int_o(aa_int_fun_int_o(P2,Xa),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_5392_groupy,axiom,
    ! [A4: assn,B4: assn,C3: assn,D4: assn,X4: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),B4)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),D4)),X4)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),B4)),C3)),D4),X4) ) ).

% groupy
tff(fact_5393_midextr,axiom,
    ! [P2: assn,Q2: assn,Q4: assn,R2: assn,X4: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Q2)),Q4)),R2),X4)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),R2)),Q2)),Q4),X4) ) ).

% midextr
tff(fact_5394_swappa,axiom,
    ! [B4: assn,A4: assn,C3: assn,X4: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B4),A4)),C3),X4)
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),B4)),C3),X4) ) ).

% swappa
tff(fact_5395_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_lis3769085826154824284t_real(set_real2,linord4252657396651189596t_real(A4)) = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_5396_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( aa_list_o_set_o(set_o2,linord3142498349692569832_set_o(A4)) = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_5397_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_list_int_set_int(set_int2,linord2612477271533052124et_int(A4)) = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_5398_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( set_Code_integer2(linord2324613341767563021nteger(A4)) = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_5399_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_list_nat_set_nat(set_nat2,linord2614967742042102400et_nat(A4)) = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_5400_ent__false__iff,axiom,
    ! [P2: assn] :
      ( entails(P2,bot_bot_assn)
    <=> ! [H4: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(P2),H4) ) ).

% ent_false_iff
tff(fact_5401_ent__trans,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( entails(P2,Q2)
     => ( entails(Q2,R2)
       => entails(P2,R2) ) ) ).

% ent_trans
tff(fact_5402_ent__refl,axiom,
    ! [P2: assn] : entails(P2,P2) ).

% ent_refl
tff(fact_5403_ent__iffI,axiom,
    ! [A4: assn,B4: assn] :
      ( entails(A4,B4)
     => ( entails(B4,A4)
       => ( A4 = B4 ) ) ) ).

% ent_iffI
tff(fact_5404_ent__star__mono,axiom,
    ! [P2: assn,P4: assn,Q2: assn,Q4: assn] :
      ( entails(P2,P4)
     => ( entails(Q2,Q4)
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Q2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P4),Q4)) ) ) ).

% ent_star_mono
tff(fact_5405_entails__def,axiom,
    ! [P2: assn,Q2: assn] :
      ( entails(P2,Q2)
    <=> ! [H4: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H4)
         => aa_Pro653082453696054616_nat_o(rep_assn(Q2),H4) ) ) ).

% entails_def
tff(fact_5406_entailsI,axiom,
    ! [P2: assn,Q2: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H)
         => aa_Pro653082453696054616_nat_o(rep_assn(Q2),H) )
     => entails(P2,Q2) ) ).

% entailsI
tff(fact_5407_entailsD,axiom,
    ! [P2: assn,Q2: assn,H2: produc3658429121746597890et_nat] :
      ( entails(P2,Q2)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H2)
       => aa_Pro653082453696054616_nat_o(rep_assn(Q2),H2) ) ) ).

% entailsD
tff(fact_5408_ent__fwd,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat,Q2: assn] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H2)
     => ( entails(P2,Q2)
       => aa_Pro653082453696054616_nat_o(rep_assn(Q2),H2) ) ) ).

% ent_fwd
tff(fact_5409_ent__false,axiom,
    ! [P2: assn] : entails(bot_bot_assn,P2) ).

% ent_false
tff(fact_5410_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( linord2612477271533052124et_int(A4) = linord2612477271533052124et_int(B4) )
     => ( aa_set_int_o(finite_finite_int,A4)
       => ( aa_set_int_o(finite_finite_int,B4)
         => ( A4 = B4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_5411_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A4: set_Code_integer,B4: set_Code_integer] :
      ( ( linord2324613341767563021nteger(A4) = linord2324613341767563021nteger(B4) )
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
         => ( A4 = B4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_5412_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( linord2614967742042102400et_nat(A4) = linord2614967742042102400et_nat(B4) )
     => ( aa_set_nat_o(finite_finite_nat,A4)
       => ( aa_set_nat_o(finite_finite_nat,B4)
         => ( A4 = B4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_5413_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A4: set_rat] : sorted_wrt_rat(ord_less_eq_rat,linord1979837681955606664et_rat(A4)) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5414_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A4: set_num] : sorted_wrt_num(ord_less_eq_num,linord8395671565052656842et_num(A4)) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5415_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A4: set_int] : sorted_wrt_int(ord_less_eq_int,linord2612477271533052124et_int(A4)) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5416_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A4: set_nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),linord2614967742042102400et_nat(A4)) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_5417_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_real] : aa_list_real_o(sorted_wrt_real(ord_less_real),linord4252657396651189596t_real(A4)) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5418_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_rat] : sorted_wrt_rat(ord_less_rat,linord1979837681955606664et_rat(A4)) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5419_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_num] : sorted_wrt_num(ord_less_num,linord8395671565052656842et_num(A4)) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5420_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_int] : sorted_wrt_int(ord_less_int,linord2612477271533052124et_int(A4)) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5421_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A4: set_nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),linord2614967742042102400et_nat(A4)) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_5422_trancl__induct2,axiom,
    ! [Ax: uint32,Ay: uint32,Bx: uint32,By: uint32,R: set_Pr3773659940955823943uint32,P2: fun_ui3478071520421546738nt32_o] :
      ( member4724532482705224080uint32(produc6884863695460953815uint32(produc1400373151660368625uint32(Ax,Ay),produc1400373151660368625uint32(Bx,By)),transi5456044692924788698uint32(R))
     => ( ! [A3: uint32,B3: uint32] :
            ( member4724532482705224080uint32(produc6884863695460953815uint32(produc1400373151660368625uint32(Ax,Ay),produc1400373151660368625uint32(A3,B3)),R)
           => aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,A3),B3) )
       => ( ! [A3: uint32,B3: uint32,Aa: uint32,Ba: uint32] :
              ( member4724532482705224080uint32(produc6884863695460953815uint32(produc1400373151660368625uint32(Ax,Ay),produc1400373151660368625uint32(A3,B3)),transi5456044692924788698uint32(R))
             => ( member4724532482705224080uint32(produc6884863695460953815uint32(produc1400373151660368625uint32(A3,B3),produc1400373151660368625uint32(Aa,Ba)),R)
               => ( aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,A3),B3)
                 => aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,Aa),Ba) ) ) )
         => aa_uint32_o(aa_uin5924812838015166427nt32_o(P2,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5423_trancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P2: fun_nat_fun_nat_o] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ax),Ay),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Bx),By)),transi243908449541399842at_nat(R))
     => ( ! [A3: nat,B3: nat] :
            ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ax),Ay),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3)),R)
           => aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3) )
       => ( ! [A3: nat,B3: nat,Aa: nat,Ba: nat] :
              ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ax),Ay),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3)),transi243908449541399842at_nat(R))
             => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Aa),Ba)),R)
               => ( aa_nat_o(aa_nat_fun_nat_o(P2,A3),B3)
                 => aa_nat_o(aa_nat_fun_nat_o(P2,Aa),Ba) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P2,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5424_trancl__induct2,axiom,
    ! [Ax: int,Ay: int,Bx: int,By: int,R: set_Pr2560585780119916871nt_int,P2: fun_int_fun_int_o] :
      ( member8566619992076573584nt_int(produc3646306378393792727nt_int(product_Pair_int_int(Ax,Ay),product_Pair_int_int(Bx,By)),transi6288783178788033498nt_int(R))
     => ( ! [A3: int,B3: int] :
            ( member8566619992076573584nt_int(produc3646306378393792727nt_int(product_Pair_int_int(Ax,Ay),product_Pair_int_int(A3,B3)),R)
           => aa_int_o(aa_int_fun_int_o(P2,A3),B3) )
       => ( ! [A3: int,B3: int,Aa: int,Ba: int] :
              ( member8566619992076573584nt_int(produc3646306378393792727nt_int(product_Pair_int_int(Ax,Ay),product_Pair_int_int(A3,B3)),transi6288783178788033498nt_int(R))
             => ( member8566619992076573584nt_int(produc3646306378393792727nt_int(product_Pair_int_int(A3,B3),product_Pair_int_int(Aa,Ba)),R)
               => ( aa_int_o(aa_int_fun_int_o(P2,A3),B3)
                 => aa_int_o(aa_int_fun_int_o(P2,Aa),Ba) ) ) )
         => aa_int_o(aa_int_fun_int_o(P2,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5425_trancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc3658429121746597890et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc3658429121746597890et_nat,R: set_Pr9066344408807579265et_nat,P2: fun_fu8419763957888848259_nat_o] :
      ( member2216856746449346506et_nat(produc8050396023069295193et_nat(produc6017135503889078682et_nat(Ax,Ay),produc6017135503889078682et_nat(Bx,By)),transi7952567189410240703et_nat(R))
     => ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat] :
            ( member2216856746449346506et_nat(produc8050396023069295193et_nat(produc6017135503889078682et_nat(Ax,Ay),produc6017135503889078682et_nat(A3,B3)),R)
           => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P2,A3),B3) )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc3658429121746597890et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc3658429121746597890et_nat] :
              ( member2216856746449346506et_nat(produc8050396023069295193et_nat(produc6017135503889078682et_nat(Ax,Ay),produc6017135503889078682et_nat(A3,B3)),transi7952567189410240703et_nat(R))
             => ( member2216856746449346506et_nat(produc8050396023069295193et_nat(produc6017135503889078682et_nat(A3,B3),produc6017135503889078682et_nat(Aa,Ba)),R)
               => ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P2,A3),B3)
                 => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P2,Aa),Ba) ) ) )
         => aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(P2,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5426_trancl__induct2,axiom,
    ! [Ax: fun_Pr6460944699146269935_nat_o,Ay: produc8799533983703829792et_nat,Bx: fun_Pr6460944699146269935_nat_o,By: produc8799533983703829792et_nat,R: set_Pr7057080801110549565et_nat,P2: fun_fu2611741329380808165_nat_o] :
      ( member2136425100634443654et_nat(produc7768440062179809557et_nat(produc7291855603632349880et_nat(Ax,Ay),produc7291855603632349880et_nat(Bx,By)),transi2167408805420631517et_nat(R))
     => ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat] :
            ( member2136425100634443654et_nat(produc7768440062179809557et_nat(produc7291855603632349880et_nat(Ax,Ay),produc7291855603632349880et_nat(A3,B3)),R)
           => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P2,A3),B3) )
       => ( ! [A3: fun_Pr6460944699146269935_nat_o,B3: produc8799533983703829792et_nat,Aa: fun_Pr6460944699146269935_nat_o,Ba: produc8799533983703829792et_nat] :
              ( member2136425100634443654et_nat(produc7768440062179809557et_nat(produc7291855603632349880et_nat(Ax,Ay),produc7291855603632349880et_nat(A3,B3)),transi2167408805420631517et_nat(R))
             => ( member2136425100634443654et_nat(produc7768440062179809557et_nat(produc7291855603632349880et_nat(A3,B3),produc7291855603632349880et_nat(Aa,Ba)),R)
               => ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P2,A3),B3)
                 => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P2,Aa),Ba) ) ) )
         => aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(P2,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5427_trancl_Ocases,axiom,
    ! [A12: uint32,A23: uint32,R: set_Pr1773385645901665561uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A12,A23),transi3114468042090999947uint32(R))
     => ( ~ member8027108493173000802uint32(produc1400373151660368625uint32(A12,A23),R)
       => ~ ! [B3: uint32] :
              ( member8027108493173000802uint32(produc1400373151660368625uint32(A12,B3),transi3114468042090999947uint32(R))
             => ~ member8027108493173000802uint32(produc1400373151660368625uint32(B3,A23),R) ) ) ) ).

% trancl.cases
tff(fact_5428_trancl_Ocases,axiom,
    ! [A12: nat,A23: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A12),A23),transi6264000038957366511cl_nat(R))
     => ( ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A12),A23),R)
       => ~ ! [B3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A12),B3),transi6264000038957366511cl_nat(R))
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),A23),R) ) ) ) ).

% trancl.cases
tff(fact_5429_trancl_Ocases,axiom,
    ! [A12: int,A23: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A12,A23),transi6261509568448316235cl_int(R))
     => ( ~ member5262025264175285858nt_int(product_Pair_int_int(A12,A23),R)
       => ~ ! [B3: int] :
              ( member5262025264175285858nt_int(product_Pair_int_int(A12,B3),transi6261509568448316235cl_int(R))
             => ~ member5262025264175285858nt_int(product_Pair_int_int(B3,A23),R) ) ) ) ).

% trancl.cases
tff(fact_5430_trancl_Osimps,axiom,
    ! [A12: uint32,A23: uint32,R: set_Pr1773385645901665561uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A12,A23),transi3114468042090999947uint32(R))
    <=> ( ? [A6: uint32,B6: uint32] :
            ( ( A12 = A6 )
            & ( A23 = B6 )
            & member8027108493173000802uint32(produc1400373151660368625uint32(A6,B6),R) )
        | ? [A6: uint32,B6: uint32,C6: uint32] :
            ( ( A12 = A6 )
            & ( A23 = C6 )
            & member8027108493173000802uint32(produc1400373151660368625uint32(A6,B6),transi3114468042090999947uint32(R))
            & member8027108493173000802uint32(produc1400373151660368625uint32(B6,C6),R) ) ) ) ).

% trancl.simps
tff(fact_5431_trancl_Osimps,axiom,
    ! [A12: nat,A23: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A12),A23),transi6264000038957366511cl_nat(R))
    <=> ( ? [A6: nat,B6: nat] :
            ( ( A12 = A6 )
            & ( A23 = B6 )
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A6),B6),R) )
        | ? [A6: nat,B6: nat,C6: nat] :
            ( ( A12 = A6 )
            & ( A23 = C6 )
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A6),B6),transi6264000038957366511cl_nat(R))
            & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B6),C6),R) ) ) ) ).

% trancl.simps
tff(fact_5432_trancl_Osimps,axiom,
    ! [A12: int,A23: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A12,A23),transi6261509568448316235cl_int(R))
    <=> ( ? [A6: int,B6: int] :
            ( ( A12 = A6 )
            & ( A23 = B6 )
            & member5262025264175285858nt_int(product_Pair_int_int(A6,B6),R) )
        | ? [A6: int,B6: int,C6: int] :
            ( ( A12 = A6 )
            & ( A23 = C6 )
            & member5262025264175285858nt_int(product_Pair_int_int(A6,B6),transi6261509568448316235cl_int(R))
            & member5262025264175285858nt_int(product_Pair_int_int(B6,C6),R) ) ) ) ).

% trancl.simps
tff(fact_5433_trancl_Or__into__trancl,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),R)
     => member8027108493173000802uint32(produc1400373151660368625uint32(A,B),transi3114468042090999947uint32(R)) ) ).

% trancl.r_into_trancl
tff(fact_5434_trancl_Or__into__trancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),R)
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),transi6264000038957366511cl_nat(R)) ) ).

% trancl.r_into_trancl
tff(fact_5435_trancl_Or__into__trancl,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),R)
     => member5262025264175285858nt_int(product_Pair_int_int(A,B),transi6261509568448316235cl_int(R)) ) ).

% trancl.r_into_trancl
tff(fact_5436_tranclE,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),transi3114468042090999947uint32(R))
     => ( ~ member8027108493173000802uint32(produc1400373151660368625uint32(A,B),R)
       => ~ ! [C: uint32] :
              ( member8027108493173000802uint32(produc1400373151660368625uint32(A,C),transi3114468042090999947uint32(R))
             => ~ member8027108493173000802uint32(produc1400373151660368625uint32(C,B),R) ) ) ) ).

% tranclE
tff(fact_5437_tranclE,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),transi6264000038957366511cl_nat(R))
     => ( ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),R)
       => ~ ! [C: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),C),transi6264000038957366511cl_nat(R))
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(C),B),R) ) ) ) ).

% tranclE
tff(fact_5438_tranclE,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),transi6261509568448316235cl_int(R))
     => ( ~ member5262025264175285858nt_int(product_Pair_int_int(A,B),R)
       => ~ ! [C: int] :
              ( member5262025264175285858nt_int(product_Pair_int_int(A,C),transi6261509568448316235cl_int(R))
             => ~ member5262025264175285858nt_int(product_Pair_int_int(C,B),R) ) ) ) ).

% tranclE
tff(fact_5439_trancl__trans,axiom,
    ! [Xa: uint32,Y: uint32,R: set_Pr1773385645901665561uint32,Z: uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y),transi3114468042090999947uint32(R))
     => ( member8027108493173000802uint32(produc1400373151660368625uint32(Y,Z),transi3114468042090999947uint32(R))
       => member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Z),transi3114468042090999947uint32(R)) ) ) ).

% trancl_trans
tff(fact_5440_trancl__trans,axiom,
    ! [Xa: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Z: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y),transi6264000038957366511cl_nat(R))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z),transi6264000038957366511cl_nat(R))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Z),transi6264000038957366511cl_nat(R)) ) ) ).

% trancl_trans
tff(fact_5441_trancl__trans,axiom,
    ! [Xa: int,Y: int,R: set_Pr958786334691620121nt_int,Z: int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Y),transi6261509568448316235cl_int(R))
     => ( member5262025264175285858nt_int(product_Pair_int_int(Y,Z),transi6261509568448316235cl_int(R))
       => member5262025264175285858nt_int(product_Pair_int_int(Xa,Z),transi6261509568448316235cl_int(R)) ) ) ).

% trancl_trans
tff(fact_5442_trancl__induct,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32,P2: fun_uint32_o] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),transi3114468042090999947uint32(R))
     => ( ! [Y3: uint32] :
            ( member8027108493173000802uint32(produc1400373151660368625uint32(A,Y3),R)
           => aa_uint32_o(P2,Y3) )
       => ( ! [Y3: uint32,Z3: uint32] :
              ( member8027108493173000802uint32(produc1400373151660368625uint32(A,Y3),transi3114468042090999947uint32(R))
             => ( member8027108493173000802uint32(produc1400373151660368625uint32(Y3,Z3),R)
               => ( aa_uint32_o(P2,Y3)
                 => aa_uint32_o(P2,Z3) ) ) )
         => aa_uint32_o(P2,B) ) ) ) ).

% trancl_induct
tff(fact_5443_trancl__induct,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,P2: fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),transi6264000038957366511cl_nat(R))
     => ( ! [Y3: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),Y3),R)
           => aa_nat_o(P2,Y3) )
       => ( ! [Y3: nat,Z3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),Y3),transi6264000038957366511cl_nat(R))
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z3),R)
               => ( aa_nat_o(P2,Y3)
                 => aa_nat_o(P2,Z3) ) ) )
         => aa_nat_o(P2,B) ) ) ) ).

% trancl_induct
tff(fact_5444_trancl__induct,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,P2: fun_int_o] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),transi6261509568448316235cl_int(R))
     => ( ! [Y3: int] :
            ( member5262025264175285858nt_int(product_Pair_int_int(A,Y3),R)
           => aa_int_o(P2,Y3) )
       => ( ! [Y3: int,Z3: int] :
              ( member5262025264175285858nt_int(product_Pair_int_int(A,Y3),transi6261509568448316235cl_int(R))
             => ( member5262025264175285858nt_int(product_Pair_int_int(Y3,Z3),R)
               => ( aa_int_o(P2,Y3)
                 => aa_int_o(P2,Z3) ) ) )
         => aa_int_o(P2,B) ) ) ) ).

% trancl_induct
tff(fact_5445_r__r__into__trancl,axiom,
    ! [A: uint32,B: uint32,R2: set_Pr1773385645901665561uint32,C2: uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),R2)
     => ( member8027108493173000802uint32(produc1400373151660368625uint32(B,C2),R2)
       => member8027108493173000802uint32(produc1400373151660368625uint32(A,C2),transi3114468042090999947uint32(R2)) ) ) ).

% r_r_into_trancl
tff(fact_5446_r__r__into__trancl,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C2: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),R2)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C2),R2)
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),C2),transi6264000038957366511cl_nat(R2)) ) ) ).

% r_r_into_trancl
tff(fact_5447_r__r__into__trancl,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,C2: int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),R2)
     => ( member5262025264175285858nt_int(product_Pair_int_int(B,C2),R2)
       => member5262025264175285858nt_int(product_Pair_int_int(A,C2),transi6261509568448316235cl_int(R2)) ) ) ).

% r_r_into_trancl
tff(fact_5448_converse__tranclE,axiom,
    ! [Xa: uint32,Z: uint32,R: set_Pr1773385645901665561uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Z),transi3114468042090999947uint32(R))
     => ( ~ member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Z),R)
       => ~ ! [Y3: uint32] :
              ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y3),R)
             => ~ member8027108493173000802uint32(produc1400373151660368625uint32(Y3,Z),transi3114468042090999947uint32(R)) ) ) ) ).

% converse_tranclE
tff(fact_5449_converse__tranclE,axiom,
    ! [Xa: nat,Z: nat,R: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Z),transi6264000038957366511cl_nat(R))
     => ( ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Z),R)
       => ~ ! [Y3: nat] :
              ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3),R)
             => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),Z),transi6264000038957366511cl_nat(R)) ) ) ) ).

% converse_tranclE
tff(fact_5450_converse__tranclE,axiom,
    ! [Xa: int,Z: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Z),transi6261509568448316235cl_int(R))
     => ( ~ member5262025264175285858nt_int(product_Pair_int_int(Xa,Z),R)
       => ~ ! [Y3: int] :
              ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Y3),R)
             => ~ member5262025264175285858nt_int(product_Pair_int_int(Y3,Z),transi6261509568448316235cl_int(R)) ) ) ) ).

% converse_tranclE
tff(fact_5451_irrefl__trancl__rD,axiom,
    ! [R: set_Pr1773385645901665561uint32,Xa: uint32,Y: uint32] :
      ( ! [X: uint32] : ~ member8027108493173000802uint32(produc1400373151660368625uint32(X,X),transi3114468042090999947uint32(R))
     => ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y),R)
       => ( Xa != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_5452_irrefl__trancl__rD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xa: nat,Y: nat] :
      ( ! [X: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X),transi6264000038957366511cl_nat(R))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y),R)
       => ( Xa != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_5453_irrefl__trancl__rD,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xa: int,Y: int] :
      ( ! [X: int] : ~ member5262025264175285858nt_int(product_Pair_int_int(X,X),transi6261509568448316235cl_int(R))
     => ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Y),R)
       => ( Xa != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_5454_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32,C2: uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),transi3114468042090999947uint32(R))
     => ( member8027108493173000802uint32(produc1400373151660368625uint32(B,C2),R)
       => member8027108493173000802uint32(produc1400373151660368625uint32(A,C2),transi3114468042090999947uint32(R)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_5455_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C2: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),transi6264000038957366511cl_nat(R))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C2),R)
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),C2),transi6264000038957366511cl_nat(R)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_5456_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,C2: int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),transi6261509568448316235cl_int(R))
     => ( member5262025264175285858nt_int(product_Pair_int_int(B,C2),R)
       => member5262025264175285858nt_int(product_Pair_int_int(A,C2),transi6261509568448316235cl_int(R)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_5457_trancl__into__trancl2,axiom,
    ! [A: uint32,B: uint32,R: set_Pr1773385645901665561uint32,C2: uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),R)
     => ( member8027108493173000802uint32(produc1400373151660368625uint32(B,C2),transi3114468042090999947uint32(R))
       => member8027108493173000802uint32(produc1400373151660368625uint32(A,C2),transi3114468042090999947uint32(R)) ) ) ).

% trancl_into_trancl2
tff(fact_5458_trancl__into__trancl2,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C2: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),R)
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),C2),transi6264000038957366511cl_nat(R))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),C2),transi6264000038957366511cl_nat(R)) ) ) ).

% trancl_into_trancl2
tff(fact_5459_trancl__into__trancl2,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,C2: int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),R)
     => ( member5262025264175285858nt_int(product_Pair_int_int(B,C2),transi6261509568448316235cl_int(R))
       => member5262025264175285858nt_int(product_Pair_int_int(A,C2),transi6261509568448316235cl_int(R)) ) ) ).

% trancl_into_trancl2
tff(fact_5460_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),Y)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Tree_is),Y))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),Y)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Tree_is),Y)))) ) ).

% local.ext
tff(fact_5461_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: produc3658429121746597890et_nat,P2: assn,R2: assn,F2: assn] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Ps),H2)
     => ( entails(P2,R2)
       => ( entails(Ps,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),F2))
         => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R2),F2)),H2) ) ) ) ).

% mod_frame_fwd
tff(fact_5462_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ).

% assnle
tff(fact_5463_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2497118539674116125T_VEBT(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5464_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2809031099982602151Ti_nat(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5465_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L886525131989349516_VEBTi(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5466_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2806540629473551875Ti_int(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5467_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_in7755481285845205148T_assn,Xs: list_int,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2018189785592951398T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2018189785592951398T_VEBT(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5468_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_int_fun_nat_assn,Xs: list_int,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8891422820522952478nt_nat(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8891422820522952478nt_nat(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5469_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_in1903134756389476099i_assn,Xs: list_int,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L114188773329725699_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L114188773329725699_VEBTi(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5470_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_int_fun_int_assn,Xs: list_int,I: nat,Xsi: list_int,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8888932350013902202nt_int(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8888932350013902202nt_int(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5471_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3204528365124325536T_VEBT(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5472_listI__assn__reinsert_H,axiom,
    ! [P2: assn,A4: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8650695023172932196BT_nat(I5,A4,Xs,Xsi)),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5473_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_VE2561185598974572172i_assn,Xa: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,Xa),Xi)),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L886525131989349516_VEBTi(I5,A4,list_u6098035379799741383_VEBTi(Xs,I,Xa),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5474_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_VE5577048379444288083T_assn,Xa: vEBT_VEBTi,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,Xa),Xi)),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2497118539674116125T_VEBT(I5,A4,list_u6098035379799741383_VEBTi(Xs,I,Xa),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5475_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_VE1571388752190420990T_assn,Xa: vEBT_VEBT,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A4,Xa),Xi)),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3204528365124325536T_VEBT(I5,A4,list_u1324408373059187874T_VEBT(Xs,I,Xa),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5476_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_re560840401387337987i_assn,Xa: real,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A4,Xa),Xi)),vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7851252805511451907_VEBTi(I5,A4,list_update_real(Xs,I,Xa),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5477_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_re8050108611700907676T_assn,Xa: real,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A4,Xa),Xi)),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3095048238742455910T_VEBT(I5,A4,list_update_real(Xs,I,Xa),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5478_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_o_9029690514724572677i_assn,Xa: $o,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A4,(Xa)),Xi)),vEBT_L6286945158656146733_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6286945158656146733_VEBTi(I5,A4,list_update_o(Xs,I,(Xa)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5479_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_o_4353168661238401050T_assn,Xa: $o,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A4,(Xa)),Xi)),vEBT_L1319876754960170684T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1319876754960170684T_VEBT(I5,A4,list_update_o(Xs,I,(Xa)),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5480_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_na6467670216208870567i_assn,Xa: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A4,Xa),Xi)),vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7489483478785760935_VEBTi(I5,A4,list_update_nat(Xs,I,Xa),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5481_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_na4489259242115493880T_assn,Xa: nat,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A4,Xa),Xi)),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8511957252848910786T_VEBT(I5,A4,list_update_nat(Xs,I,Xa),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5482_listI__assn__reinsert__upd,axiom,
    ! [P2: assn,A4: fun_VE905575822219544865i_assn,Xa: vEBT_VEBT,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,Q2: assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(A4,Xa),Xi)),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(I5,A4,list_u1324408373059187874T_VEBT(Xs,I,Xa),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5483_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Tree_is),I))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% recomp
tff(fact_5484_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Tree_is),I))),Rest)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Rest),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% repack
tff(fact_5485_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) ).

% finite_atLeastLessThan
tff(fact_5486_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I),set_or3540276404033026485et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),I)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_5487_atLeastLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( aa_set_real_o(member_real(I),set_or66887138388493659n_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_5488_atLeastLessThan__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or8585797421378605585et_int(L,U))
    <=> ( aa_set_int_o(ord_less_eq_set_int(L),I)
        & aa_set_int_o(ord_less_set_int(I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_5489_atLeastLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( aa_set_rat_o(member_rat(I),set_or4029947393144176647an_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_5490_atLeastLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( aa_set_num_o(member_num(I),set_or1222409239386451017an_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_5491_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_or4665077453230672383an_nat(L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_5492_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I),set_or4662586982721622107an_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_5493_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),set_or8404916559141939852nteger(L,U))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),I)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_5494_atLeastLessThan__empty,axiom,
    ! [B: $o,A: $o] :
      ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,(B)),(A))
     => ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
tff(fact_5495_atLeastLessThan__empty,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B),A)
     => ( set_or8585797421378605585et_int(A,B) = bot_bot_set_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_5496_atLeastLessThan__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
tff(fact_5497_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
tff(fact_5498_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_5499_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_5500_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
tff(fact_5501_atLeastLessThan__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o = set_or7139685690850216873Than_o((A),(B)) )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5502_atLeastLessThan__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or66887138388493659n_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5503_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or4029947393144176647an_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5504_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or1222409239386451017an_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5505_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5506_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or4662586982721622107an_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5507_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or8404916559141939852nteger(A,B) )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5508_atLeastLessThan__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( set_or7139685690850216873Than_o((A),(B)) = bot_bot_set_o )
    <=> ~ aa_o_o(aa_o_fun_o_o(ord_less_o,(A)),(B)) ) ).

% atLeastLessThan_empty_iff
tff(fact_5509_atLeastLessThan__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or66887138388493659n_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_5510_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_5511_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or1222409239386451017an_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_5512_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_5513_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or4662586982721622107an_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_5514_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = bot_bo3990330152332043303nteger )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% atLeastLessThan_empty_iff
tff(fact_5515_infinite__Ico__iff,axiom,
    ! [A: real,B: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or66887138388493659n_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Ico_iff
tff(fact_5516_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or4029947393144176647an_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Ico_iff
tff(fact_5517_ivl__subset,axiom,
    ! [I: rat,J: rat,M: rat,Na: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or4029947393144176647an_rat(I,J)),set_or4029947393144176647an_rat(M,Na))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,J),I)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),I)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,J),Na) ) ) ) ).

% ivl_subset
tff(fact_5518_ivl__subset,axiom,
    ! [I: num,J: num,M: num,Na: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or1222409239386451017an_num(I,J)),set_or1222409239386451017an_num(M,Na))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,J),I)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),I)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,J),Na) ) ) ) ).

% ivl_subset
tff(fact_5519_ivl__subset,axiom,
    ! [I: nat,J: nat,M: nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),Na) ) ) ) ).

% ivl_subset
tff(fact_5520_ivl__subset,axiom,
    ! [I: int,J: int,M: int,Na: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or4662586982721622107an_int(I,J)),set_or4662586982721622107an_int(M,Na))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),I)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),Na) ) ) ) ).

% ivl_subset
tff(fact_5521_ivl__subset,axiom,
    ! [I: code_integer,J: code_integer,M: code_integer,Na: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(I,J)),set_or8404916559141939852nteger(M,Na))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,J),I)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),I)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,J),Na) ) ) ) ).

% ivl_subset
tff(fact_5522_ivl__diff,axiom,
    ! [I: rat,Na: rat,M: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),Na)
     => ( minus_minus_set_rat(set_or4029947393144176647an_rat(I,M),set_or4029947393144176647an_rat(I,Na)) = set_or4029947393144176647an_rat(Na,M) ) ) ).

% ivl_diff
tff(fact_5523_ivl__diff,axiom,
    ! [I: num,Na: num,M: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),Na)
     => ( minus_minus_set_num(set_or1222409239386451017an_num(I,M),set_or1222409239386451017an_num(I,Na)) = set_or1222409239386451017an_num(Na,M) ) ) ).

% ivl_diff
tff(fact_5524_ivl__diff,axiom,
    ! [I: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Na)
     => ( minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),M),aa_nat_set_nat(set_or4665077453230672383an_nat(I),Na)) = aa_nat_set_nat(set_or4665077453230672383an_nat(Na),M) ) ) ).

% ivl_diff
tff(fact_5525_ivl__diff,axiom,
    ! [I: int,Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),Na)
     => ( minus_minus_set_int(set_or4662586982721622107an_int(I,M),set_or4662586982721622107an_int(I,Na)) = set_or4662586982721622107an_int(Na,M) ) ) ).

% ivl_diff
tff(fact_5526_ivl__diff,axiom,
    ! [I: code_integer,Na: code_integer,M: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I),Na)
     => ( minus_2355218937544613996nteger(set_or8404916559141939852nteger(I,M),set_or8404916559141939852nteger(I,Na)) = set_or8404916559141939852nteger(Na,M) ) ) ).

% ivl_diff
tff(fact_5527_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xa: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Xs)),I)
     => ( list_u6098035379799741383_VEBTi(Xs,I,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_5528_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_5529_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),I)
     => ( list_update_real(Xs,I,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_5530_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I)
     => ( list_update_o(Xs,I,(Xa)) = Xs ) ) ).

% list_update_beyond
tff(fact_5531_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),I)
     => ( list_update_nat(Xs,I,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_5532_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : ( aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ) ).

% image_Suc_atLeastLessThan
tff(fact_5533_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_VEBT_VEBT,Y: vEBT_VEBT,Xa: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_s6755466524823107622T_VEBT(TreeList3) )
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),Xa)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xa)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xa))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xa)))) ) ) ).

% tcd
tff(fact_5534_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_real,Y: vEBT_VEBT,Xa: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_real(TreeList3) )
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),Xa)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xa)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xa))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xa)))) ) ) ).

% tcd
tff(fact_5535_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_o,Y: vEBT_VEBT,Xa: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = size_size_list_o(TreeList3) )
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),Xa)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xa)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xa))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xa)))) ) ) ).

% tcd
tff(fact_5536_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList3: list_nat,Y: vEBT_VEBT,Xa: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_list_nat_nat(size_size_list_nat,TreeList3) )
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Y),Xa)),snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xa)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xa))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,I,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,I,Y),list_u6098035379799741383_VEBTi(Tree_is,I,Xa)))) ) ) ).

% tcd
tff(fact_5537_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,Xa: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_nat_int(nth_int(list_update_int(L,J,Xa)),I) = aa_nat_int(nth_int(L),I) ) ) ).

% nth_update_invalid
tff(fact_5538_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J: nat,Xa: vEBT_VEBTi] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,J,Xa)),I) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I) ) ) ).

% nth_update_invalid
tff(fact_5539_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J: nat,Xa: vEBT_VEBT] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,J,Xa)),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I) ) ) ).

% nth_update_invalid
tff(fact_5540_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J: nat,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_nat_real(nth_real(list_update_real(L,J,Xa)),I) = aa_nat_real(nth_real(L),I) ) ) ).

% nth_update_invalid
tff(fact_5541_nth__update__invalid,axiom,
    ! [I: nat,L: list_o,J: nat,Xa: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_nat_o(nth_o(list_update_o(L,J,(Xa))),I)
      <=> aa_nat_o(nth_o(L),I) ) ) ).

% nth_update_invalid
tff(fact_5542_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,Xa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L))
     => ( aa_nat_nat(nth_nat(list_update_nat(L,J,Xa)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_update_invalid
tff(fact_5543_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_5544_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xa: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_5545_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_5546_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_5547_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xa))),I)
      <=> (Xa) ) ) ).

% nth_list_update_eq
tff(fact_5548_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_5549_atLeastLessThan__singleton,axiom,
    ! [M: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = aa_set_nat_set_nat(insert_nat(M),bot_bot_set_nat) ) ).

% atLeastLessThan_singleton
tff(fact_5550_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),Y)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Tree_is),Y))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% txe
tff(fact_5551_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( aa_list_int_set_int(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))) = aa_list_int_set_int(set_int2,Xs) ) ) ) ).

% set_swap
tff(fact_5552_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s7982070591426661849_VEBTi(Xs))
       => ( set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),J)),J,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I))) = set_VEBT_VEBTi2(Xs) ) ) ) ).

% set_swap
tff(fact_5553_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J)),J,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I))) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs) ) ) ) ).

% set_swap
tff(fact_5554_set__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(Xs))
       => ( aa_lis3769085826154824284t_real(set_real2,list_update_real(list_update_real(Xs,I,aa_nat_real(nth_real(Xs),J)),J,aa_nat_real(nth_real(Xs),I))) = aa_lis3769085826154824284t_real(set_real2,Xs) ) ) ) ).

% set_swap
tff(fact_5555_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
       => ( aa_list_o_set_o(set_o2,list_update_o(list_update_o(Xs,I,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I))) = aa_list_o_set_o(set_o2,Xs) ) ) ) ).

% set_swap
tff(fact_5556_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_list_nat_set_nat(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))) = aa_list_nat_set_nat(set_nat2,Xs) ) ) ) ).

% set_swap
tff(fact_5557_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_uint32,M: nat,Na: nat] :
      ( groups833757482993574392uint32(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),zero_zero_uint32,aa_uint32_uint32(plus_plus_uint32(groups833757482993574392uint32(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))),aa_nat_uint32(G,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_5558_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,Na: nat] :
      ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))),aa_nat_rat(G,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_5559_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,Na: nat] :
      ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))),aa_nat_int(G,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_5560_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] :
      ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))),aa_nat_nat(G,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_5561_sum_Oop__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] :
      ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))),aa_nat_real(G,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_5562_nth__image__indices,axiom,
    ! [L: list_VEBT_VEBTi] : ( image_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(L))) = set_VEBT_VEBTi2(L) ) ).

% nth_image_indices
tff(fact_5563_nth__image__indices,axiom,
    ! [L: list_set_nat] : ( aa_set7246812951530217472et_nat(image_nat_set_nat(nth_set_nat(L)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s3254054031482475050et_nat(L))) = set_set_nat2(L) ) ).

% nth_image_indices
tff(fact_5564_nth__image__indices,axiom,
    ! [L: list_int] : ( aa_set_nat_set_int(image_nat_int(nth_int(L)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_int(L))) = aa_list_int_set_int(set_int2,L) ) ).

% nth_image_indices
tff(fact_5565_nth__image__indices,axiom,
    ! [L: list_VEBT_VEBT] : ( image_nat_VEBT_VEBT(nth_VEBT_VEBT(L),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(L))) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L) ) ).

% nth_image_indices
tff(fact_5566_nth__image__indices,axiom,
    ! [L: list_real] : ( image_nat_real(nth_real(L),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(L))) = aa_lis3769085826154824284t_real(set_real2,L) ) ).

% nth_image_indices
tff(fact_5567_nth__image__indices,axiom,
    ! [L: list_o] : ( image_nat_o(nth_o(L),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_o(L))) = aa_list_o_set_o(set_o2,L) ) ).

% nth_image_indices
tff(fact_5568_nth__image__indices,axiom,
    ! [L: list_nat] : ( aa_set_nat_set_nat(image_nat_nat(nth_nat(L)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_list_nat_nat(size_size_list_nat,L))) = aa_list_nat_set_nat(set_nat2,L) ) ).

% nth_image_indices
tff(fact_5569_deleti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_V1365221501068881998eletei(T,Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa))) ) ).

% deleti'_rf_abstr
tff(fact_5570_atLeastLessThan__eq__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
       => ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C2,D) )
        <=> ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5571_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
       => ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C2,D) )
        <=> ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5572_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),D)
       => ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C2,D) )
        <=> ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5573_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
       => ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D) )
        <=> ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5574_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
       => ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C2,D) )
        <=> ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5575_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D)
       => ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C2,D) )
        <=> ( ( A = C2 )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5576_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C2,D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5577_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C2,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5578_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C2,D) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),D)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5579_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5580_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C2,D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5581_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C2,D) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D)
         => ( A = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5582_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( set_or66887138388493659n_real(A,B) = set_or66887138388493659n_real(C2,D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5583_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( set_or4029947393144176647an_rat(A,B) = set_or4029947393144176647an_rat(C2,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5584_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( set_or1222409239386451017an_num(A,B) = set_or1222409239386451017an_num(C2,D) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5585_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A),B) = aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5586_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( set_or4662586982721622107an_int(A,B) = set_or4662586982721622107an_int(C2,D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5587_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] :
      ( ( set_or8404916559141939852nteger(A,B) = set_or8404916559141939852nteger(C2,D) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D)
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5588_listI__assn__conv,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,A4: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi] :
      ( ( Na = size_s6755466524823107622T_VEBT(Xs) )
     => ( vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na),A4,Xs,Xsi) = vEBT_L6296928887356842470_VEBTi(A4,Xs,Xsi) ) ) ).

% listI_assn_conv
tff(fact_5589_list__assn__conv__idx,axiom,
    ! [A4: fun_VE905575822219544865i_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] : ( vEBT_L6296928887356842470_VEBTi(A4,Xs,Xsi) = vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),A4,Xs,Xsi) ) ).

% list_assn_conv_idx
tff(fact_5590_list__update__code_I3_J,axiom,
    ! [Xa: int,Xs: list_int,I: nat,Y: int] : ( list_update_int(cons_int(Xa,Xs),aa_nat_nat(suc,I),Y) = cons_int(Xa,list_update_int(Xs,I,Y)) ) ).

% list_update_code(3)
tff(fact_5591_list__update__code_I3_J,axiom,
    ! [Xa: nat,Xs: list_nat,I: nat,Y: nat] : ( list_update_nat(cons_nat(Xa,Xs),aa_nat_nat(suc,I),Y) = cons_nat(Xa,list_update_nat(Xs,I,Y)) ) ).

% list_update_code(3)
tff(fact_5592_list__update__code_I3_J,axiom,
    ! [Xa: $o,Xs: list_o,I: nat,Y: $o] : ( list_update_o(cons_o((Xa),Xs),aa_nat_nat(suc,I),(Y)) = cons_o((Xa),list_update_o(Xs,I,(Y))) ) ).

% list_update_code(3)
tff(fact_5593_list__update__code_I3_J,axiom,
    ! [Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] : ( list_u6098035379799741383_VEBTi(cons_VEBT_VEBTi(Xa,Xs),aa_nat_nat(suc,I),Y) = cons_VEBT_VEBTi(Xa,list_u6098035379799741383_VEBTi(Xs,I,Y)) ) ).

% list_update_code(3)
tff(fact_5594_list__update__code_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] : ( list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(Xa,Xs),aa_nat_nat(suc,I),Y) = cons_VEBT_VEBT(Xa,list_u1324408373059187874T_VEBT(Xs,I,Y)) ) ).

% list_update_code(3)
tff(fact_5595_list__update__code_I2_J,axiom,
    ! [Xa: int,Xs: list_int,Y: int] : ( list_update_int(cons_int(Xa,Xs),zero_zero_nat,Y) = cons_int(Y,Xs) ) ).

% list_update_code(2)
tff(fact_5596_list__update__code_I2_J,axiom,
    ! [Xa: nat,Xs: list_nat,Y: nat] : ( list_update_nat(cons_nat(Xa,Xs),zero_zero_nat,Y) = cons_nat(Y,Xs) ) ).

% list_update_code(2)
tff(fact_5597_list__update__code_I2_J,axiom,
    ! [Xa: $o,Xs: list_o,Y: $o] : ( list_update_o(cons_o((Xa),Xs),zero_zero_nat,(Y)) = cons_o((Y),Xs) ) ).

% list_update_code(2)
tff(fact_5598_list__update__code_I2_J,axiom,
    ! [Xa: vEBT_VEBTi,Xs: list_VEBT_VEBTi,Y: vEBT_VEBTi] : ( list_u6098035379799741383_VEBTi(cons_VEBT_VEBTi(Xa,Xs),zero_zero_nat,Y) = cons_VEBT_VEBTi(Y,Xs) ) ).

% list_update_code(2)
tff(fact_5599_list__update__code_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] : ( list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(Xa,Xs),zero_zero_nat,Y) = cons_VEBT_VEBT(Y,Xs) ) ).

% list_update_code(2)
tff(fact_5600_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A4: set_set_nat,Xa: set_nat,I: nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Xs)),A4)
     => ( aa_set_set_nat_o(member_set_nat(Xa),A4)
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I,Xa))),A4) ) ) ).

% set_update_subsetI
tff(fact_5601_set__update__subsetI,axiom,
    ! [Xs: list_nat,A4: set_nat,Xa: nat,I: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Xs)),A4)
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,list_update_nat(Xs,I,Xa))),A4) ) ) ).

% set_update_subsetI
tff(fact_5602_set__update__subsetI,axiom,
    ! [Xs: list_real,A4: set_real,Xa: real,I: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Xs)),A4)
     => ( aa_set_real_o(member_real(Xa),A4)
       => aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,list_update_real(Xs,I,Xa))),A4) ) ) ).

% set_update_subsetI
tff(fact_5603_set__update__subsetI,axiom,
    ! [Xs: list_o,A4: set_o,Xa: $o,I: nat] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Xs)),A4)
     => ( aa_set_o_o(member_o((Xa)),A4)
       => aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,list_update_o(Xs,I,(Xa)))),A4) ) ) ).

% set_update_subsetI
tff(fact_5604_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBTi,A4: set_VEBT_VEBTi,Xa: vEBT_VEBTi,I: nat] :
      ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Xs),A4)
     => ( member_VEBT_VEBTi(Xa,A4)
       => ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,Xa)),A4) ) ) ).

% set_update_subsetI
tff(fact_5605_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A4: set_VEBT_VEBT,Xa: vEBT_VEBT,I: nat] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)),A4)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
       => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(Xs,I,Xa))),A4) ) ) ).

% set_update_subsetI
tff(fact_5606_set__update__subsetI,axiom,
    ! [Xs: list_int,A4: set_int,Xa: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Xs)),A4)
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,list_update_int(Xs,I,Xa))),A4) ) ) ).

% set_update_subsetI
tff(fact_5607_infinite__Ico,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_set_real_o(finite_finite_real,set_or66887138388493659n_real(A,B)) ) ).

% infinite_Ico
tff(fact_5608_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_set_rat_o(finite_finite_rat,set_or4029947393144176647an_rat(A,B)) ) ).

% infinite_Ico
tff(fact_5609_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A,B)),set_or4029947393144176647an_rat(C2,D))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5610_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or1222409239386451017an_num(A,B)),set_or1222409239386451017an_num(C2,D))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5611_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5612_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or4662586982721622107an_int(A,B)),set_or4662586982721622107an_int(C2,D))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5613_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(A,B)),set_or8404916559141939852nteger(C2,D))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),D) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5614_ex__nat__less__eq,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Na)
          & aa_nat_o(P2,M5) )
    <=> ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
          & aa_nat_o(P2,X5) ) ) ).

% ex_nat_less_eq
tff(fact_5615_all__nat__less__eq,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Na)
         => aa_nat_o(P2,M5) )
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
         => aa_nat_o(P2,X5) ) ) ).

% all_nat_less_eq
tff(fact_5616_atLeastLessThan0,axiom,
    ! [M: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ) ).

% atLeastLessThan0
tff(fact_5617_listI__assn__conv_H,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,A4: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi,F2: assn] :
      ( ( Na = size_s6755466524823107622T_VEBT(Xs) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na),A4,Xs,Xsi)),F2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6296928887356842470_VEBTi(A4,Xs,Xsi)),F2) ) ) ).

% listI_assn_conv'
tff(fact_5618_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N6),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
     => aa_set_nat_o(finite_finite_nat,N6) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_5619_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A4))))
     => ( A4 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_5620_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1279224858307276611T_VEBT(A4,Xs,Xsi)),H2)
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5621_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_re8050108611700907676T_assn,Xs: list_real,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L4595930785310033027T_VEBT(A4,Xs,Xsi)),H2)
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5622_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_o_4353168661238401050T_assn,Xs: list_o,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1750719106661372127T_VEBT(A4,Xs,Xsi)),H2)
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5623_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_na4489259242115493880T_assn,Xs: list_nat,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L8158188754432654943T_VEBT(A4,Xs,Xsi)),H2)
     => ( size_s6755466524823107622T_VEBT(Xsi) = aa_list_nat_nat(size_size_list_nat,Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5624_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L5781919052683127133T_real(A4,Xs,Xsi)),H2)
     => ( size_size_list_real(Xsi) = size_s6755466524823107622T_VEBT(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5625_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_re6303261851654292474l_assn,Xs: list_real,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1930518968523514909l_real(A4,Xs,Xsi)),H2)
     => ( size_size_list_real(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5626_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_o_fun_real_assn,Xs: list_o,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L4725278957065240257o_real(A4,Xs,Xsi)),H2)
     => ( size_size_list_real(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5627_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_na448437756743965086l_assn,Xs: list_nat,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L6102073776069194049t_real(A4,Xs,Xsi)),H2)
     => ( size_size_list_real(Xsi) = aa_list_nat_nat(size_size_list_nat,Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5628_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_VE2104784059010244834o_assn,Xs: list_VEBT_VEBT,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L7489408758114837031VEBT_o(A4,Xs,Xsi)),H2)
     => ( size_size_list_o(Xsi) = size_s6755466524823107622T_VEBT(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5629_extract__pre__list__assn__lengthD,axiom,
    ! [A4: fun_real_fun_o_assn,Xs: list_real,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L6234343332106409831real_o(A4,Xs,Xsi)),H2)
     => ( size_size_list_o(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5630_set__update__memI,axiom,
    ! [Na: nat,Xs: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,list_update_int(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_5631_set__update__memI,axiom,
    ! [Na: nat,Xs: list_set_nat,Xa: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
     => aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(list_update_set_nat(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_5632_set__update__memI,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBTi,Xa: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_5633_set__update__memI,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_5634_set__update__memI,axiom,
    ! [Na: nat,Xs: list_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,list_update_real(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_5635_set__update__memI,axiom,
    ! [Na: nat,Xs: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,list_update_o(Xs,Na,(Xa)))) ) ).

% set_update_memI
tff(fact_5636_set__update__memI,axiom,
    ! [Na: nat,Xs: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
     => aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,list_update_nat(Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_5637_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,Xa: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,list_update_int(L,I,Y)))
      <=> ( ( Xa = Y )
          | ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,L))
            & ! [Y5: int] : aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,list_update_int(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_5638_in__set__upd__eq,axiom,
    ! [I: nat,L: list_set_nat,Xa: set_nat,Y: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3254054031482475050et_nat(L))
     => ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(list_update_set_nat(L,I,Y)))
      <=> ( ( Xa = Y )
          | ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(L))
            & ! [Y5: set_nat] : aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(list_update_set_nat(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_5639_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xa: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( Xa = Y )
          | ( member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(L))
            & ! [Y5: vEBT_VEBTi] : member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_5640_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xa: vEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( Xa = Y )
          | ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L))
            & ! [Y5: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_5641_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,list_update_real(L,I,Y)))
      <=> ( ( Xa = Y )
          | ( aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,L))
            & ! [Y5: real] : aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,list_update_real(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_5642_in__set__upd__eq,axiom,
    ! [I: nat,L: list_o,Xa: $o,Y: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,list_update_o(L,I,(Y))))
      <=> ( ( (Xa)
          <=> (Y) )
          | ( aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,L))
            & ! [Y5: $o] : aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,list_update_o(L,I,(Y5)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_5643_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L))
     => ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y)))
      <=> ( ( Xa = Y )
          | ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,L))
            & ! [Y5: nat] : aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_5644_in__set__upd__cases,axiom,
    ! [Xa: int,L: list_int,I: nat,Y: int] :
      ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,list_update_int(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
         => ( Xa != Y ) )
       => aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,L)) ) ) ).

% in_set_upd_cases
tff(fact_5645_in__set__upd__cases,axiom,
    ! [Xa: set_nat,L: list_set_nat,I: nat,Y: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(list_update_set_nat(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3254054031482475050et_nat(L))
         => ( Xa != Y ) )
       => aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(L)) ) ) ).

% in_set_upd_cases
tff(fact_5646_in__set__upd__cases,axiom,
    ! [Xa: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
         => ( Xa != Y ) )
       => member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(L)) ) ) ).

% in_set_upd_cases
tff(fact_5647_in__set__upd__cases,axiom,
    ! [Xa: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
         => ( Xa != Y ) )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L)) ) ) ).

% in_set_upd_cases
tff(fact_5648_in__set__upd__cases,axiom,
    ! [Xa: real,L: list_real,I: nat,Y: real] :
      ( aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,list_update_real(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
         => ( Xa != Y ) )
       => aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,L)) ) ) ).

% in_set_upd_cases
tff(fact_5649_in__set__upd__cases,axiom,
    ! [Xa: $o,L: list_o,I: nat,Y: $o] :
      ( aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,list_update_o(L,I,(Y))))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
         => ( (Xa)
          <=> ~ (Y) ) )
       => aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,L)) ) ) ).

% in_set_upd_cases
tff(fact_5650_in__set__upd__cases,axiom,
    ! [Xa: nat,L: list_nat,I: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L))
         => ( Xa != Y ) )
       => aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,L)) ) ) ).

% in_set_upd_cases
tff(fact_5651_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,Xa: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,list_update_int(L,I,Y)))
      <=> ( ( Xa = Y )
          | ! [Y5: int] : aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,list_update_int(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_5652_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_set_nat,Xa: set_nat,Y: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3254054031482475050et_nat(L))
     => ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(list_update_set_nat(L,I,Y)))
      <=> ( ( Xa = Y )
          | ! [Y5: set_nat] : aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(list_update_set_nat(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_5653_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xa: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( Xa = Y )
          | ! [Y5: vEBT_VEBTi] : member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_5654_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xa: vEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( Xa = Y )
          | ! [Y5: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_5655_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,list_update_real(L,I,Y)))
      <=> ( ( Xa = Y )
          | ! [Y5: real] : aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,list_update_real(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_5656_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_o,Xa: $o,Y: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,list_update_o(L,I,(Y))))
      <=> ( ( (Xa)
          <=> (Y) )
          | ! [Y5: $o] : aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,list_update_o(L,I,(Y5)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_5657_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L))
     => ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y)))
      <=> ( ( Xa = Y )
          | ! [Y5: nat] : aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_5658_nth__list__update_H,axiom,
    ! [L: list_int,I: nat,Xa: int,J: nat] :
      ( aa_nat_int(nth_int(list_update_int(L,I,Xa)),J) = $ite(
          ( ( I = J )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L)) ),
          Xa,
          aa_nat_int(nth_int(L),J) ) ) ).

% nth_list_update'
tff(fact_5659_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBTi,I: nat,Xa: vEBT_VEBTi,J: nat] :
      ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,Xa)),J) = $ite(
          ( ( I = J )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)) ),
          Xa,
          aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),J) ) ) ).

% nth_list_update'
tff(fact_5660_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBT,I: nat,Xa: vEBT_VEBT,J: nat] :
      ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,Xa)),J) = $ite(
          ( ( I = J )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)) ),
          Xa,
          aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),J) ) ) ).

% nth_list_update'
tff(fact_5661_nth__list__update_H,axiom,
    ! [L: list_real,I: nat,Xa: real,J: nat] :
      ( aa_nat_real(nth_real(list_update_real(L,I,Xa)),J) = $ite(
          ( ( I = J )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L)) ),
          Xa,
          aa_nat_real(nth_real(L),J) ) ) ).

% nth_list_update'
tff(fact_5662_nth__list__update_H,axiom,
    ! [L: list_o,I: nat,Xa: $o,J: nat] :
      ( aa_nat_o(nth_o(list_update_o(L,I,(Xa))),J)
    <=> $ite(
          ( ( I = J )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L)) ),
          (Xa),
          aa_nat_o(nth_o(L),J) ) ) ).

% nth_list_update'
tff(fact_5663_nth__list__update_H,axiom,
    ! [L: list_nat,I: nat,Xa: nat,J: nat] :
      ( aa_nat_nat(nth_nat(list_update_nat(L,I,Xa)),J) = $ite(
          ( ( I = J )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L)) ),
          Xa,
          aa_nat_nat(nth_nat(L),J) ) ) ).

% nth_list_update'
tff(fact_5664_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,I,Xa) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_5665_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xa: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( ( list_u6098035379799741383_VEBTi(Xs,I,Xa) = Xs )
      <=> ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_5666_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,Xa) = Xs )
      <=> ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_5667_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( ( list_update_real(Xs,I,Xa) = Xs )
      <=> ( aa_nat_real(nth_real(Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_5668_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I,(Xa)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I)
        <=> (Xa) ) ) ) ).

% list_update_same_conv
tff(fact_5669_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( ( list_update_nat(Xs,I,Xa) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_5670_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,Xa: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xa)),J) = $ite(I = J,Xa,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_5671_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xa: vEBT_VEBTi,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xa)),J) = $ite(I = J,Xa,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),J)) ) ) ).

% nth_list_update
tff(fact_5672_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xa: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xa)),J) = $ite(I = J,Xa,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J)) ) ) ).

% nth_list_update
tff(fact_5673_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,Xa: real,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xa)),J) = $ite(I = J,Xa,aa_nat_real(nth_real(Xs),J)) ) ) ).

% nth_list_update
tff(fact_5674_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,Xa: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xa))),J)
      <=> $ite(I = J,(Xa),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_5675_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,Xa: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xa)),J) = $ite(I = J,Xa,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_5676_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,Xa: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,list_update_nat(Xs,I,Xa))),aa_set_nat_set_nat(insert_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs))) ).

% set_update_subset_insert
tff(fact_5677_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,Xa: real] : aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,list_update_real(Xs,I,Xa))),aa_set_real_set_real(insert_real(Xa),aa_lis3769085826154824284t_real(set_real2,Xs))) ).

% set_update_subset_insert
tff(fact_5678_set__update__subset__insert,axiom,
    ! [Xs: list_o,I: nat,Xa: $o] : aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,list_update_o(Xs,I,(Xa)))),aa_set_o_set_o(insert_o((Xa)),aa_list_o_set_o(set_o2,Xs))) ).

% set_update_subset_insert
tff(fact_5679_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xa: vEBT_VEBTi] : ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,Xa)),insert_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(Xs))) ).

% set_update_subset_insert
tff(fact_5680_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xa: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(Xs,I,Xa))),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))) ).

% set_update_subset_insert
tff(fact_5681_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,Xa: int] : aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,list_update_int(Xs,I,Xa))),aa_set_int_set_int(insert_int(Xa),aa_list_int_set_int(set_int2,Xs))) ).

% set_update_subset_insert
tff(fact_5682_sum_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat,G: fun_nat_nat,H2: fun_nat_nat] :
      ( ( A = C2 )
     => ( ( B = D )
       => ( ! [X: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),D)
               => ( aa_nat_nat(G,X) = aa_nat_nat(H2,X) ) ) )
         => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = aa_set_nat_nat(groups3542108847815614940at_nat(H2),aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_5683_sum_Oivl__cong,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat,G: fun_nat_real,H2: fun_nat_real] :
      ( ( A = C2 )
     => ( ( B = D )
       => ( ! [X: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),D)
               => ( aa_nat_real(G,X) = aa_nat_real(H2,X) ) ) )
         => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B)) = groups6591440286371151544t_real(H2,aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_5684_sum_Oivl__cong,axiom,
    ! [A: int,C2: int,B: int,D: int,G: fun_int_int,H2: fun_int_int] :
      ( ( A = C2 )
     => ( ( B = D )
       => ( ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),X)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),D)
               => ( aa_int_int(G,X) = aa_int_int(H2,X) ) ) )
         => ( groups4538972089207619220nt_int(G,set_or4662586982721622107an_int(A,B)) = groups4538972089207619220nt_int(H2,set_or4662586982721622107an_int(C2,D)) ) ) ) ) ).

% sum.ivl_cong
tff(fact_5685_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,Na: nat,P3: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),P3)
       => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(Na),P3))) = groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_5686_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,Na: nat,P3: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),P3)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(Na),P3))) = aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_5687_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,Na: nat,P3: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),P3)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(Na),P3))) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_5688_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,Na: nat,P3: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),P3)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(Na),P3))) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_5689_sum__diff__nat__ivl,axiom,
    ! [M: nat,Na: nat,P3: nat,F: fun_nat_uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),P3)
       => ( minus_minus_uint32(groups833757482993574392uint32(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3)),groups833757482993574392uint32(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))) = groups833757482993574392uint32(F,aa_nat_set_nat(set_or4665077453230672383an_nat(Na),P3)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_5690_sum__diff__nat__ivl,axiom,
    ! [M: nat,Na: nat,P3: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),P3)
       => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3))),aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or4665077453230672383an_nat(Na),P3)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_5691_sum__diff__nat__ivl,axiom,
    ! [M: nat,Na: nat,P3: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),P3)
       => ( aa_real_real(minus_minus_real(groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),P3))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na))) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(Na),P3)) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_5692_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE2561185598974572172i_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L886525131989349516_VEBTi(I5,A4,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L886525131989349516_VEBTi(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5693_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE5577048379444288083T_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2497118539674116125T_VEBT(I5,A4,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = vEBT_L2497118539674116125T_VEBT(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5694_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L3204528365124325536T_VEBT(I5,A4,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = vEBT_L3204528365124325536T_VEBT(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5695_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A4: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L7851252805511451907_VEBTi(I5,A4,list_update_real(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L7851252805511451907_VEBTi(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5696_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A4: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L3095048238742455910T_VEBT(I5,A4,list_update_real(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = vEBT_L3095048238742455910T_VEBT(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5697_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A4: fun_o_9029690514724572677i_assn,X1: $o,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L6286945158656146733_VEBTi(I5,A4,list_update_o(Xs,I,(X1)),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L6286945158656146733_VEBTi(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5698_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A4: fun_o_4353168661238401050T_assn,X1: $o,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L1319876754960170684T_VEBT(I5,A4,list_update_o(Xs,I,(X1)),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = vEBT_L1319876754960170684T_VEBT(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5699_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A4: fun_na6467670216208870567i_assn,X1: nat,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( vEBT_L7489483478785760935_VEBTi(I5,A4,list_update_nat(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L7489483478785760935_VEBTi(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5700_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A4: fun_na4489259242115493880T_assn,X1: nat,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( vEBT_L8511957252848910786T_VEBT(I5,A4,list_update_nat(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = vEBT_L8511957252848910786T_VEBT(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5701_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: fun_VE905575822219544865i_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L1528199826722428489_VEBTi(I5,A4,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = vEBT_L1528199826722428489_VEBTi(I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5702_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = aa_set_nat_set_nat(insert_nat(Na),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)) ) ).

% atLeast0_lessThan_Suc
tff(fact_5703_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N6: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N6),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N6)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_5704_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_VE2561185598974572172i_assn,Xa: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,Xa),Xi)),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L886525131989349516_VEBTi(I5,A4,list_u6098035379799741383_VEBTi(Xs,I,Xa),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5705_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_VE5577048379444288083T_assn,Xa: vEBT_VEBTi,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,Xa),Xi)),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L2497118539674116125T_VEBT(I5,A4,list_u6098035379799741383_VEBTi(Xs,I,Xa),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5706_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_VE1571388752190420990T_assn,Xa: vEBT_VEBT,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A4,Xa),Xi)),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3204528365124325536T_VEBT(I5,A4,list_u1324408373059187874T_VEBT(Xs,I,Xa),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5707_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_re560840401387337987i_assn,Xa: real,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A4,Xa),Xi)),vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7851252805511451907_VEBTi(I5,A4,list_update_real(Xs,I,Xa),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5708_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_re8050108611700907676T_assn,Xa: real,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A4,Xa),Xi)),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L3095048238742455910T_VEBT(I5,A4,list_update_real(Xs,I,Xa),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5709_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_o_9029690514724572677i_assn,Xa: $o,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A4,(Xa)),Xi)),vEBT_L6286945158656146733_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6286945158656146733_VEBTi(I5,A4,list_update_o(Xs,I,(Xa)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5710_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_o_4353168661238401050T_assn,Xa: $o,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A4,(Xa)),Xi)),vEBT_L1319876754960170684T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1319876754960170684T_VEBT(I5,A4,list_update_o(Xs,I,(Xa)),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5711_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_na6467670216208870567i_assn,Xa: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A4,Xa),Xi)),vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7489483478785760935_VEBTi(I5,A4,list_update_nat(Xs,I,Xa),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5712_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_na4489259242115493880T_assn,Xa: nat,Xi: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A4,Xa),Xi)),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8511957252848910786T_VEBT(I5,A4,list_update_nat(Xs,I,Xa),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5713_listI__assn__reinsert__upd_H,axiom,
    ! [P2: assn,A4: fun_VE2561185598974572172i_assn,Xa: vEBT_VEBTi,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_Time_Heap_o,Q2: fun_o_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,Xa),Xi)),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_hoare_triple_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L886525131989349516_VEBTi(I5,A4,list_u6098035379799741383_VEBTi(Xs,I,Xa),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_hoare_triple_o(P2,C2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5714_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_uint32,K: nat] :
      ( ( aa_nat_uint32(F,zero_zero_nat) = zero_zero_uint32 )
     => ( groups833757482993574392uint32(F,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups833757482993574392uint32(F,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_5715_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_5716_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_int,K: nat] :
      ( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
     => ( aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_int(groups3539618377306564664at_int(F),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_5717_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_5718_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_5719_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))),aa_nat_rat(G,Na)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_5720_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))),aa_nat_int(G,Na)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_5721_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))),aa_nat_nat(G,Na)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_5722_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))),aa_nat_real(G,Na)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_5723_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),Na))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_5724_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),Na))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_5725_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),Na))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_5726_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),Na))) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_5727_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_rat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_5728_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_int(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_5729_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_nat(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_5730_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),aa_nat_nat(suc,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B))),aa_nat_real(G,B)) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_5731_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),aa_set_nat_set_nat(insert_nat(Na),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)),bot_bot_set_nat) ) ).

% atLeastLessThanSuc
tff(fact_5732_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_5733_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,Xa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_set_int(insert_int(aa_nat_int(nth_int(L),I)),aa_list_int_set_int(set_int2,list_update_int(L,I,Xa))) = aa_set_int_set_int(insert_int(Xa),aa_list_int_set_int(set_int2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_5734_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xa: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( insert_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),I),set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Xa))) = insert_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_5735_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(L,I,Xa))) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_5736_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_set_real(insert_real(aa_nat_real(nth_real(L),I)),aa_lis3769085826154824284t_real(set_real2,list_update_real(L,I,Xa))) = aa_set_real_set_real(insert_real(Xa),aa_lis3769085826154824284t_real(set_real2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_5737_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,Xa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_set_o_set_o(insert_o(aa_nat_o(nth_o(L),I)),aa_list_o_set_o(set_o2,list_update_o(L,I,(Xa)))) = aa_set_o_set_o(insert_o((Xa)),aa_list_o_set_o(set_o2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_5738_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L))
     => ( aa_set_nat_set_nat(insert_nat(aa_nat_nat(nth_nat(L),I)),aa_list_nat_set_nat(set_nat2,list_update_nat(L,I,Xa))) = aa_set_nat_set_nat(insert_nat(Xa),aa_list_nat_set_nat(set_nat2,L)) ) ) ).

% insert_swap_set_eq
tff(fact_5739_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE2561185598974572172i_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L886525131989349516_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,X1),X2)),vEBT_L886525131989349516_VEBTi(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5740_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A4: fun_VE5577048379444288083T_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2497118539674116125T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,X1),X2)),vEBT_L2497118539674116125T_VEBT(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5741_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L3204528365124325536T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A4,X1),X2)),vEBT_L3204528365124325536T_VEBT(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5742_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A4: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L7851252805511451907_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_update_real(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A4,X1),X2)),vEBT_L7851252805511451907_VEBTi(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5743_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A4: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L3095048238742455910T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_update_real(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A4,X1),X2)),vEBT_L3095048238742455910T_VEBT(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5744_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A4: fun_o_9029690514724572677i_assn,X1: $o,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L6286945158656146733_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_update_o(Xs,I,(X1)),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A4,(X1)),X2)),vEBT_L6286945158656146733_VEBTi(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5745_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A4: fun_o_4353168661238401050T_assn,X1: $o,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L1319876754960170684T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_update_o(Xs,I,(X1)),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A4,(X1)),X2)),vEBT_L1319876754960170684T_VEBT(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5746_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A4: fun_na6467670216208870567i_assn,X1: nat,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( vEBT_L7489483478785760935_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_update_nat(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A4,X1),X2)),vEBT_L7489483478785760935_VEBTi(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5747_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A4: fun_na4489259242115493880T_assn,X1: nat,Xsi: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( vEBT_L8511957252848910786T_VEBT(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_update_nat(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A4,X1),X2)),vEBT_L8511957252848910786T_VEBT(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5748_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A4: fun_VE905575822219544865i_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L1528199826722428489_VEBTi(aa_set_nat_set_nat(insert_nat(I),I5),A4,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u6098035379799741383_VEBTi(Xsi,I,X2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(A4,X1),X2)),vEBT_L1528199826722428489_VEBTi(I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5749_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,Xa: vEBT_VEBTi,Xb: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H2),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( Xaa = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L) )
       => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xaa),Xa)),aa_o_assn(pure_assn,Xb = vEBT_vebt_mint(Xaa)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H2,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(H2),bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H2,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_o_assn(pure_assn,Xb = vEBT_vebt_mint(Xaa)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H2,Xaa),list_u6098035379799741383_VEBTi(Tree_is,H2,Xa)))) ) ) ).

% big_assn_simp'
tff(fact_5750_big__assn__simp,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,L: nat,Xa: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H2),size_s6755466524823107622T_VEBT(TreeList))
     => entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L)),Xa)),aa_o_assn(pure_assn,Xaa = vEBT_vebt_mint(aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L))))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H2,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),aa_set_nat_set_nat(insert_nat(H2),bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H2,Xa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),X14))),aa_o_assn(pure_assn,Xaa = vEBT_vebt_mint(aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L))))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H2,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L)),list_u6098035379799741383_VEBTi(Tree_is,H2,Xa)))) ) ).

% big_assn_simp
tff(fact_5751_listI__assn__wrap__insert,axiom,
    ! [P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu2),Uua)),Xi)),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(I5,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu2),Uua)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5752_listI__assn__wrap__insert,axiom,
    ! [P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_Time_Heap_o,Q2: fun_o_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu2),Uua)),Xi)),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_hoare_triple_o(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(I5,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu2),Uua)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_hoare_triple_o(P2,C2,Q2) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5753_listI__assn__wrap__insert,axiom,
    ! [P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_T2636463487746394924on_nat,Q2: fun_option_nat_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu2),Uua)),Xi)),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_7629718768684598413on_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(I5,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu2),Uua)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_7629718768684598413on_nat(P2,C2,Q2) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5754_listI__assn__wrap__insert,axiom,
    ! [P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F2: assn,C2: heap_Time_Heap_nat,Q2: fun_nat_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu2),Uua)),Xi)),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I5,aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs,Xsi))),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( hoare_3067605981109127869le_nat(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1528199826722428489_VEBTi(I5,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu2),Uua)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F2),C2,Q2)
           => hoare_3067605981109127869le_nat(P2,C2,Q2) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5755_set__update__distinct,axiom,
    ! [Xs: list_VEBT_VEBTi,Na: nat,Xa: vEBT_VEBTi] :
      ( distinct_VEBT_VEBTi(Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s7982070591426661849_VEBTi(Xs))
       => ( set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,Na,Xa)) = insert_VEBT_VEBTi(Xa,minus_3697805406911847364_VEBTi(set_VEBT_VEBTi2(Xs),insert_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Na),bot_bo8982466882572371071_VEBTi))) ) ) ) ).

% set_update_distinct
tff(fact_5756_set__update__distinct,axiom,
    ! [Xs: list_VEBT_VEBT,Na: nat,Xa: vEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,list_u1324408373059187874T_VEBT(Xs,Na,Xa)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),minus_5127226145743854075T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Na)),bot_bo8194388402131092736T_VEBT))) ) ) ) ).

% set_update_distinct
tff(fact_5757_set__update__distinct,axiom,
    ! [Xs: list_real,Na: nat,Xa: real] :
      ( aa_list_real_o(distinct_real,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
       => ( aa_lis3769085826154824284t_real(set_real2,list_update_real(Xs,Na,Xa)) = aa_set_real_set_real(insert_real(Xa),minus_minus_set_real(aa_lis3769085826154824284t_real(set_real2,Xs),aa_set_real_set_real(insert_real(aa_nat_real(nth_real(Xs),Na)),bot_bot_set_real))) ) ) ) ).

% set_update_distinct
tff(fact_5758_set__update__distinct,axiom,
    ! [Xs: list_int,Na: nat,Xa: int] :
      ( distinct_int(Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
       => ( aa_list_int_set_int(set_int2,list_update_int(Xs,Na,Xa)) = aa_set_int_set_int(insert_int(Xa),minus_minus_set_int(aa_list_int_set_int(set_int2,Xs),aa_set_int_set_int(insert_int(aa_nat_int(nth_int(Xs),Na)),bot_bot_set_int))) ) ) ) ).

% set_update_distinct
tff(fact_5759_set__update__distinct,axiom,
    ! [Xs: list_o,Na: nat,Xa: $o] :
      ( aa_list_o_o(distinct_o,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
       => ( aa_list_o_set_o(set_o2,list_update_o(Xs,Na,(Xa))) = aa_set_o_set_o(insert_o((Xa)),minus_minus_set_o(aa_list_o_set_o(set_o2,Xs),aa_set_o_set_o(insert_o(aa_nat_o(nth_o(Xs),Na)),bot_bot_set_o))) ) ) ) ).

% set_update_distinct
tff(fact_5760_set__update__distinct,axiom,
    ! [Xs: list_nat,Na: nat,Xa: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_list_nat_set_nat(set_nat2,list_update_nat(Xs,Na,Xa)) = aa_set_nat_set_nat(insert_nat(Xa),minus_minus_set_nat(aa_list_nat_set_nat(set_nat2,Xs),aa_set_nat_set_nat(insert_nat(aa_nat_nat(nth_nat(Xs),Na)),bot_bot_set_nat))) ) ) ) ).

% set_update_distinct
tff(fact_5761_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H)
         => hoare_1429296392585015714_VEBTi(P2,C2,Q2) )
     => hoare_1429296392585015714_VEBTi(P2,C2,Q2) ) ).

% hoare_triple_preI
tff(fact_5762_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_Time_Heap_o,Q2: fun_o_assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H)
         => hoare_hoare_triple_o(P2,C2,Q2) )
     => hoare_hoare_triple_o(P2,C2,Q2) ) ).

% hoare_triple_preI
tff(fact_5763_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_T2636463487746394924on_nat,Q2: fun_option_nat_assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H)
         => hoare_7629718768684598413on_nat(P2,C2,Q2) )
     => hoare_7629718768684598413on_nat(P2,C2,Q2) ) ).

% hoare_triple_preI
tff(fact_5764_hoare__triple__preI,axiom,
    ! [P2: assn,C2: heap_Time_Heap_nat,Q2: fun_nat_assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H)
         => hoare_3067605981109127869le_nat(P2,C2,Q2) )
     => hoare_3067605981109127869le_nat(P2,C2,Q2) ) ).

% hoare_triple_preI
tff(fact_5765_nth__image,axiom,
    ! [L: nat,Xs: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),size_s7982070591426661849_VEBTi(Xs))
     => ( image_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),L)) = set_VEBT_VEBTi2(take_VEBT_VEBTi(L,Xs)) ) ) ).

% nth_image
tff(fact_5766_nth__image,axiom,
    ! [L: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),size_s3254054031482475050et_nat(Xs))
     => ( aa_set7246812951530217472et_nat(image_nat_set_nat(nth_set_nat(Xs)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),L)) = set_set_nat2(take_set_nat(L,Xs)) ) ) ).

% nth_image
tff(fact_5767_nth__image,axiom,
    ! [L: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),size_size_list_int(Xs))
     => ( aa_set_nat_set_int(image_nat_int(nth_int(Xs)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),L)) = aa_list_int_set_int(set_int2,take_int(L,Xs)) ) ) ).

% nth_image
tff(fact_5768_nth__image,axiom,
    ! [L: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),size_s6755466524823107622T_VEBT(Xs))
     => ( image_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),L)) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(L,Xs)) ) ) ).

% nth_image
tff(fact_5769_nth__image,axiom,
    ! [L: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),size_size_list_real(Xs))
     => ( image_nat_real(nth_real(Xs),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),L)) = aa_lis3769085826154824284t_real(set_real2,take_real(L,Xs)) ) ) ).

% nth_image
tff(fact_5770_nth__image,axiom,
    ! [L: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),size_size_list_o(Xs))
     => ( image_nat_o(nth_o(Xs),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),L)) = aa_list_o_set_o(set_o2,take_o(L,Xs)) ) ) ).

% nth_image
tff(fact_5771_nth__image,axiom,
    ! [L: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_set_nat_set_nat(image_nat_nat(nth_nat(Xs)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),L)) = aa_list_nat_set_nat(set_nat2,take_nat(L,Xs)) ) ) ).

% nth_image
tff(fact_5772_pure__assn__eq__conv,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( aa_o_assn(pure_assn,(P2)) = aa_o_assn(pure_assn,(Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% pure_assn_eq_conv
tff(fact_5773_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,(A))),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
          ( (A)
          & (B) )) ) ).

% merge_pure_star
tff(fact_5774_take__update,axiom,
    ! [Na: nat,L: list_nat,I: nat,Xa: nat] : ( take_nat(Na,list_update_nat(L,I,Xa)) = list_update_nat(take_nat(Na,L),I,Xa) ) ).

% take_update
tff(fact_5775_take__update,axiom,
    ! [Na: nat,L: list_VEBT_VEBTi,I: nat,Xa: vEBT_VEBTi] : ( take_VEBT_VEBTi(Na,list_u6098035379799741383_VEBTi(L,I,Xa)) = list_u6098035379799741383_VEBTi(take_VEBT_VEBTi(Na,L),I,Xa) ) ).

% take_update
tff(fact_5776_take__update,axiom,
    ! [Na: nat,L: list_VEBT_VEBT,I: nat,Xa: vEBT_VEBT] : ( take_VEBT_VEBT(Na,list_u1324408373059187874T_VEBT(L,I,Xa)) = list_u1324408373059187874T_VEBT(take_VEBT_VEBT(Na,L),I,Xa) ) ).

% take_update
tff(fact_5777_pure__assn__eq__emp__iff,axiom,
    ! [P2: $o] :
      ( ( aa_o_assn(pure_assn,(P2)) = one_one_assn )
    <=> (P2) ) ).

% pure_assn_eq_emp_iff
tff(fact_5778_pure__true,axiom,
    aa_o_assn(pure_assn,$true) = one_one_assn ).

% pure_true
tff(fact_5779_pure__false,axiom,
    aa_o_assn(pure_assn,$false) = bot_bot_assn ).

% pure_false
tff(fact_5780_pure__assn__eq__false__iff,axiom,
    ! [P2: $o] :
      ( ( aa_o_assn(pure_assn,(P2)) = bot_bot_assn )
    <=> ~ (P2) ) ).

% pure_assn_eq_false_iff
tff(fact_5781_take__Suc__Cons,axiom,
    ! [Na: nat,Xa: int,Xs: list_int] : ( take_int(aa_nat_nat(suc,Na),cons_int(Xa,Xs)) = cons_int(Xa,take_int(Na,Xs)) ) ).

% take_Suc_Cons
tff(fact_5782_take__Suc__Cons,axiom,
    ! [Na: nat,Xa: nat,Xs: list_nat] : ( take_nat(aa_nat_nat(suc,Na),cons_nat(Xa,Xs)) = cons_nat(Xa,take_nat(Na,Xs)) ) ).

% take_Suc_Cons
tff(fact_5783_take__Suc__Cons,axiom,
    ! [Na: nat,Xa: $o,Xs: list_o] : ( take_o(aa_nat_nat(suc,Na),cons_o((Xa),Xs)) = cons_o((Xa),take_o(Na,Xs)) ) ).

% take_Suc_Cons
tff(fact_5784_take__all,axiom,
    ! [Xs: list_VEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),Na)
     => ( take_VEBT_VEBT(Na,Xs) = Xs ) ) ).

% take_all
tff(fact_5785_take__all,axiom,
    ! [Xs: list_real,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),Na)
     => ( take_real(Na,Xs) = Xs ) ) ).

% take_all
tff(fact_5786_take__all,axiom,
    ! [Xs: list_o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),Na)
     => ( take_o(Na,Xs) = Xs ) ) ).

% take_all
tff(fact_5787_take__all,axiom,
    ! [Xs: list_nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),Na)
     => ( take_nat(Na,Xs) = Xs ) ) ).

% take_all
tff(fact_5788_take__all__iff,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( ( take_VEBT_VEBT(Na,Xs) = Xs )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),Na) ) ).

% take_all_iff
tff(fact_5789_take__all__iff,axiom,
    ! [Na: nat,Xs: list_real] :
      ( ( take_real(Na,Xs) = Xs )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),Na) ) ).

% take_all_iff
tff(fact_5790_take__all__iff,axiom,
    ! [Na: nat,Xs: list_o] :
      ( ( take_o(Na,Xs) = Xs )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),Na) ) ).

% take_all_iff
tff(fact_5791_take__all__iff,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( ( take_nat(Na,Xs) = Xs )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),Na) ) ).

% take_all_iff
tff(fact_5792_nth__take,axiom,
    ! [I: nat,Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(take_VEBT_VEBT(Na,Xs)),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I) ) ) ).

% nth_take
tff(fact_5793_nth__take,axiom,
    ! [I: nat,Na: nat,Xs: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(take_VEBT_VEBTi(Na,Xs)),I) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I) ) ) ).

% nth_take
tff(fact_5794_nth__take,axiom,
    ! [I: nat,Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_int(nth_int(take_int(Na,Xs)),I) = aa_nat_int(nth_int(Xs),I) ) ) ).

% nth_take
tff(fact_5795_nth__take,axiom,
    ! [I: nat,Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_nat(nth_nat(take_nat(Na,Xs)),I) = aa_nat_nat(nth_nat(Xs),I) ) ) ).

% nth_take
tff(fact_5796_take__update__cancel,axiom,
    ! [Na: nat,M: nat,Xs: list_nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( take_nat(Na,list_update_nat(Xs,M,Y)) = take_nat(Na,Xs) ) ) ).

% take_update_cancel
tff(fact_5797_take__update__cancel,axiom,
    ! [Na: nat,M: nat,Xs: list_VEBT_VEBTi,Y: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( take_VEBT_VEBTi(Na,list_u6098035379799741383_VEBTi(Xs,M,Y)) = take_VEBT_VEBTi(Na,Xs) ) ) ).

% take_update_cancel
tff(fact_5798_take__update__cancel,axiom,
    ! [Na: nat,M: nat,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( take_VEBT_VEBT(Na,list_u1324408373059187874T_VEBT(Xs,M,Y)) = take_VEBT_VEBT(Na,Xs) ) ) ).

% take_update_cancel
tff(fact_5799_ent__pure__pre__iff,axiom,
    ! [P2: assn,B: $o,Q2: assn] :
      ( entails(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),aa_o_assn(pure_assn,(B))),Q2)
    <=> ( (B)
       => entails(P2,Q2) ) ) ).

% ent_pure_pre_iff
tff(fact_5800_mod__pure__star__dist,axiom,
    ! [P2: assn,B: $o,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),aa_o_assn(pure_assn,(B)))),H2)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H2)
        & (B) ) ) ).

% mod_pure_star_dist
tff(fact_5801_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q2: assn] :
      ( entails(aa_o_assn(pure_assn,(B)),Q2)
    <=> ( (B)
       => entails(one_one_assn,Q2) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_5802_ent__pure__post__iff,axiom,
    ! [P2: assn,Q2: assn,B: $o] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q2),aa_o_assn(pure_assn,(B))))
    <=> ( ! [H4: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H4)
           => (B) )
        & entails(P2,Q2) ) ) ).

% ent_pure_post_iff
tff(fact_5803_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_o_assn(pure_assn,(B))),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> (B) ) ).

% mod_h_bot_iff(1)
tff(fact_5804_ent__pure__post__iff__sng,axiom,
    ! [P2: assn,B: $o] :
      ( entails(P2,aa_o_assn(pure_assn,(B)))
    <=> ( ! [H4: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H4)
           => (B) )
        & entails(P2,one_one_assn) ) ) ).

% ent_pure_post_iff_sng
tff(fact_5805_distinct__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( distinct_int(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I)))
        <=> distinct_int(Xs) ) ) ) ).

% distinct_swap
tff(fact_5806_distinct__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s7982070591426661849_VEBTi(Xs))
       => ( distinct_VEBT_VEBTi(list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),J)),J,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)))
        <=> distinct_VEBT_VEBTi(Xs) ) ) ) ).

% distinct_swap
tff(fact_5807_distinct__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J)),J,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)))
        <=> aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs) ) ) ) ).

% distinct_swap
tff(fact_5808_distinct__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(Xs))
       => ( aa_list_real_o(distinct_real,list_update_real(list_update_real(Xs,I,aa_nat_real(nth_real(Xs),J)),J,aa_nat_real(nth_real(Xs),I)))
        <=> aa_list_real_o(distinct_real,Xs) ) ) ) ).

% distinct_swap
tff(fact_5809_distinct__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
       => ( aa_list_o_o(distinct_o,list_update_o(list_update_o(Xs,I,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I)))
        <=> aa_list_o_o(distinct_o,Xs) ) ) ) ).

% distinct_swap
tff(fact_5810_distinct__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_list_nat_o(distinct_nat,list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I)))
        <=> aa_list_nat_o(distinct_nat,Xs) ) ) ) ).

% distinct_swap
tff(fact_5811_VEBT__internal_OperInsTrans_Ointros_I2_J,axiom,
    ! [T: vEBT_VEBT,Sa: vEBT_VEBT,Xa: nat,U: vEBT_VEBT] :
      ( ( T = aa_nat_VEBT_VEBT(vEBT_vebt_insert(Sa),Xa) )
     => ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,T),U)
       => aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,Sa),U) ) ) ).

% VEBT_internal.perInsTrans.intros(2)
tff(fact_5812_VEBT__internal_OperInsTrans_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,A12),A23)
     => ( ( A23 != A12 )
       => ~ ! [T6: vEBT_VEBT] :
              ( ? [X: nat] : ( T6 = aa_nat_VEBT_VEBT(vEBT_vebt_insert(A12),X) )
             => ~ aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,T6),A23) ) ) ) ).

% VEBT_internal.perInsTrans.cases
tff(fact_5813_VEBT__internal_OperInsTrans_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,A12),A23)
    <=> ( ? [T3: vEBT_VEBT] :
            ( ( A12 = T3 )
            & ( A23 = T3 ) )
        | ? [T3: vEBT_VEBT,S6: vEBT_VEBT,X5: nat,U2: vEBT_VEBT] :
            ( ( A12 = S6 )
            & ( A23 = U2 )
            & ( T3 = aa_nat_VEBT_VEBT(vEBT_vebt_insert(S6),X5) )
            & aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(vEBT_V6289311342943941716sTrans,T3),U2) ) ) ) ).

% VEBT_internal.perInsTrans.simps
tff(fact_5814_set__take__subset,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(Na,Xs))),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)) ).

% set_take_subset
tff(fact_5815_set__take__subset,axiom,
    ! [Na: nat,Xs: list_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,take_nat(Na,Xs))),aa_list_nat_set_nat(set_nat2,Xs)) ).

% set_take_subset
tff(fact_5816_set__take__subset,axiom,
    ! [Na: nat,Xs: list_real] : aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,take_real(Na,Xs))),aa_lis3769085826154824284t_real(set_real2,Xs)) ).

% set_take_subset
tff(fact_5817_set__take__subset,axiom,
    ! [Na: nat,Xs: list_o] : aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,take_o(Na,Xs))),aa_list_o_set_o(set_o2,Xs)) ).

% set_take_subset
tff(fact_5818_set__take__subset,axiom,
    ! [Na: nat,Xs: list_int] : aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,take_int(Na,Xs))),aa_list_int_set_int(set_int2,Xs)) ).

% set_take_subset
tff(fact_5819_sorted__take,axiom,
    ! [Xs: list_rat,Na: nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => sorted_wrt_rat(ord_less_eq_rat,take_rat(Na,Xs)) ) ).

% sorted_take
tff(fact_5820_sorted__take,axiom,
    ! [Xs: list_num,Na: nat] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => sorted_wrt_num(ord_less_eq_num,take_num(Na,Xs)) ) ).

% sorted_take
tff(fact_5821_sorted__take,axiom,
    ! [Xs: list_nat,Na: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),take_nat(Na,Xs)) ) ).

% sorted_take
tff(fact_5822_sorted__take,axiom,
    ! [Xs: list_int,Na: nat] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
     => sorted_wrt_int(ord_less_eq_int,take_int(Na,Xs)) ) ).

% sorted_take
tff(fact_5823_finite__distinct__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ? [Xs2: list_VEBT_VEBT] :
          ( ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs2) = A4 )
          & aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs2) ) ) ).

% finite_distinct_list
tff(fact_5824_finite__distinct__list,axiom,
    ! [A4: set_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ? [Xs2: list_real] :
          ( ( aa_lis3769085826154824284t_real(set_real2,Xs2) = A4 )
          & aa_list_real_o(distinct_real,Xs2) ) ) ).

% finite_distinct_list
tff(fact_5825_finite__distinct__list,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ? [Xs2: list_o] :
          ( ( aa_list_o_set_o(set_o2,Xs2) = A4 )
          & aa_list_o_o(distinct_o,Xs2) ) ) ).

% finite_distinct_list
tff(fact_5826_finite__distinct__list,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ? [Xs2: list_nat] :
          ( ( aa_list_nat_set_nat(set_nat2,Xs2) = A4 )
          & aa_list_nat_o(distinct_nat,Xs2) ) ) ).

% finite_distinct_list
tff(fact_5827_finite__distinct__list,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ? [Xs2: list_int] :
          ( ( aa_list_int_set_int(set_int2,Xs2) = A4 )
          & distinct_int(Xs2) ) ) ).

% finite_distinct_list
tff(fact_5828_finite__distinct__list,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ? [Xs2: list_complex] :
          ( ( set_complex2(Xs2) = A4 )
          & distinct_complex(Xs2) ) ) ).

% finite_distinct_list
tff(fact_5829_finite__distinct__list,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ? [Xs2: list_Code_integer] :
          ( ( set_Code_integer2(Xs2) = A4 )
          & distin1543349897113766820nteger(Xs2) ) ) ).

% finite_distinct_list
tff(fact_5830_finite__set__image,axiom,
    ! [A4: set_list_VEBT_VEBT] :
      ( finite9070685783930458813T_VEBT(image_6463372868993444447T_VEBT(set_VEBT_VEBT2,A4))
     => ( ! [Xs2: list_VEBT_VEBT] :
            ( member2936631157270082147T_VEBT(Xs2,A4)
           => aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs2) )
       => finite3004134309566078307T_VEBT(A4) ) ) ).

% finite_set_image
tff(fact_5831_finite__set__image,axiom,
    ! [A4: set_list_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,image_1775855109352712557et_nat(set_nat2,A4))
     => ( ! [Xs2: list_nat] :
            ( member_list_nat(Xs2,A4)
           => aa_list_nat_o(distinct_nat,Xs2) )
       => finite8100373058378681591st_nat(A4) ) ) ).

% finite_set_image
tff(fact_5832_finite__set__image,axiom,
    ! [A4: set_list_real] :
      ( finite9007344921179782393t_real(image_6239767680843085477t_real(set_real2,A4))
     => ( ! [Xs2: list_real] :
            ( member_list_real(Xs2,A4)
           => aa_list_real_o(distinct_real,Xs2) )
       => finite306553202115118035t_real(A4) ) ) ).

% finite_set_image
tff(fact_5833_finite__set__image,axiom,
    ! [A4: set_list_o] :
      ( finite_finite_set_o(image_list_o_set_o(set_o2,A4))
     => ( ! [Xs2: list_o] :
            ( member_list_o(Xs2,A4)
           => aa_list_o_o(distinct_o,Xs2) )
       => finite_finite_list_o(A4) ) ) ).

% finite_set_image
tff(fact_5834_set__take__subset__set__take,axiom,
    ! [M: nat,Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(M,Xs))),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,take_VEBT_VEBT(Na,Xs))) ) ).

% set_take_subset_set_take
tff(fact_5835_set__take__subset__set__take,axiom,
    ! [M: nat,Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,take_nat(M,Xs))),aa_list_nat_set_nat(set_nat2,take_nat(Na,Xs))) ) ).

% set_take_subset_set_take
tff(fact_5836_set__take__subset__set__take,axiom,
    ! [M: nat,Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,take_real(M,Xs))),aa_lis3769085826154824284t_real(set_real2,take_real(Na,Xs))) ) ).

% set_take_subset_set_take
tff(fact_5837_set__take__subset__set__take,axiom,
    ! [M: nat,Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,take_o(M,Xs))),aa_list_o_set_o(set_o2,take_o(Na,Xs))) ) ).

% set_take_subset_set_take
tff(fact_5838_set__take__subset__set__take,axiom,
    ! [M: nat,Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,take_int(M,Xs))),aa_list_int_set_int(set_int2,take_int(Na,Xs))) ) ).

% set_take_subset_set_take
tff(fact_5839_strict__sorted__iff,axiom,
    ! [L: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_real),L)
    <=> ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),L)
        & aa_list_real_o(distinct_real,L) ) ) ).

% strict_sorted_iff
tff(fact_5840_strict__sorted__iff,axiom,
    ! [L: list_rat] :
      ( sorted_wrt_rat(ord_less_rat,L)
    <=> ( sorted_wrt_rat(ord_less_eq_rat,L)
        & distinct_rat(L) ) ) ).

% strict_sorted_iff
tff(fact_5841_strict__sorted__iff,axiom,
    ! [L: list_num] :
      ( sorted_wrt_num(ord_less_num,L)
    <=> ( sorted_wrt_num(ord_less_eq_num,L)
        & distinct_num(L) ) ) ).

% strict_sorted_iff
tff(fact_5842_strict__sorted__iff,axiom,
    ! [L: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),L)
    <=> ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),L)
        & aa_list_nat_o(distinct_nat,L) ) ) ).

% strict_sorted_iff
tff(fact_5843_strict__sorted__iff,axiom,
    ! [L: list_int] :
      ( sorted_wrt_int(ord_less_int,L)
    <=> ( sorted_wrt_int(ord_less_eq_int,L)
        & distinct_int(L) ) ) ).

% strict_sorted_iff
tff(fact_5844_distinct__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBTi] :
      ( distinct_VEBT_VEBTi(Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s7982070591426661849_VEBTi(Xs))
         => ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_s7982070591426661849_VEBTi(Xs))
             => ( ( I3 != J3 )
               => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I3) != aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_5845_distinct__conv__nth,axiom,
    ! [Xs: list_int] :
      ( distinct_int(Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
         => ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_int(Xs))
             => ( ( I3 != J3 )
               => ( aa_nat_int(nth_int(Xs),I3) != aa_nat_int(nth_int(Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_5846_distinct__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
         => ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_s6755466524823107622T_VEBT(Xs))
             => ( ( I3 != J3 )
               => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3) != aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_5847_distinct__conv__nth,axiom,
    ! [Xs: list_real] :
      ( aa_list_real_o(distinct_real,Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
         => ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_real(Xs))
             => ( ( I3 != J3 )
               => ( aa_nat_real(nth_real(Xs),I3) != aa_nat_real(nth_real(Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_5848_distinct__conv__nth,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(distinct_o,Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
         => ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),size_size_list_o(Xs))
             => ( ( I3 != J3 )
               => ~ ( aa_nat_o(nth_o(Xs),I3)
                  <=> aa_nat_o(nth_o(Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_5849_distinct__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_list_nat_nat(size_size_list_nat,Xs))
         => ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),aa_list_nat_nat(size_size_list_nat,Xs))
             => ( ( I3 != J3 )
               => ( aa_nat_nat(nth_nat(Xs),I3) != aa_nat_nat(nth_nat(Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_5850_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,J: nat] :
      ( distinct_VEBT_VEBTi(Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s7982070591426661849_VEBTi(Xs))
         => ( ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_5851_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( distinct_int(Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
         => ( ( aa_nat_int(nth_int(Xs),I) = aa_nat_int(nth_int(Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_5852_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,J: nat] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
         => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_5853_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_real,I: nat,J: nat] :
      ( aa_list_real_o(distinct_real,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(Xs))
         => ( ( aa_nat_real(nth_real(Xs),I) = aa_nat_real(nth_real(Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_5854_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_o,I: nat,J: nat] :
      ( aa_list_o_o(distinct_o,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
         => ( ( aa_nat_o(nth_o(Xs),I)
            <=> aa_nat_o(nth_o(Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_5855_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,Xs))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,Xs))
         => ( ( aa_nat_nat(nth_nat(Xs),I) = aa_nat_nat(nth_nat(Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_5856_distinct__length__le,axiom,
    ! [Ys: list_VEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Ys)
     => ( ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys) = aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Ys)),size_s6755466524823107622T_VEBT(Xs)) ) ) ).

% distinct_length_le
tff(fact_5857_distinct__length__le,axiom,
    ! [Ys: list_real,Xs: list_real] :
      ( aa_list_real_o(distinct_real,Ys)
     => ( ( aa_lis3769085826154824284t_real(set_real2,Ys) = aa_lis3769085826154824284t_real(set_real2,Xs) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Ys)),size_size_list_real(Xs)) ) ) ).

% distinct_length_le
tff(fact_5858_distinct__length__le,axiom,
    ! [Ys: list_o,Xs: list_o] :
      ( aa_list_o_o(distinct_o,Ys)
     => ( ( aa_list_o_set_o(set_o2,Ys) = aa_list_o_set_o(set_o2,Xs) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Ys)),size_size_list_o(Xs)) ) ) ).

% distinct_length_le
tff(fact_5859_distinct__length__le,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( aa_list_nat_o(distinct_nat,Ys)
     => ( ( aa_list_nat_set_nat(set_nat2,Ys) = aa_list_nat_set_nat(set_nat2,Xs) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Ys)),aa_list_nat_nat(size_size_list_nat,Xs)) ) ) ).

% distinct_length_le
tff(fact_5860_sorted__distinct__set__unique,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Xs)
     => ( aa_list_real_o(distinct_real,Xs)
       => ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Ys)
         => ( aa_list_real_o(distinct_real,Ys)
           => ( ( aa_lis3769085826154824284t_real(set_real2,Xs) = aa_lis3769085826154824284t_real(set_real2,Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_5861_sorted__distinct__set__unique,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
     => ( aa_list_o_o(distinct_o,Xs)
       => ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Ys)
         => ( aa_list_o_o(distinct_o,Ys)
           => ( ( aa_list_o_set_o(set_o2,Xs) = aa_list_o_set_o(set_o2,Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_5862_sorted__distinct__set__unique,axiom,
    ! [Xs: list_rat,Ys: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => ( distinct_rat(Xs)
       => ( sorted_wrt_rat(ord_less_eq_rat,Ys)
         => ( distinct_rat(Ys)
           => ( ( set_rat2(Xs) = set_rat2(Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_5863_sorted__distinct__set__unique,axiom,
    ! [Xs: list_num,Ys: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => ( distinct_num(Xs)
       => ( sorted_wrt_num(ord_less_eq_num,Ys)
         => ( distinct_num(Ys)
           => ( ( set_num2(Xs) = set_num2(Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_5864_sorted__distinct__set__unique,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
     => ( aa_list_nat_o(distinct_nat,Xs)
       => ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Ys)
         => ( aa_list_nat_o(distinct_nat,Ys)
           => ( ( aa_list_nat_set_nat(set_nat2,Xs) = aa_list_nat_set_nat(set_nat2,Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_5865_sorted__distinct__set__unique,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
     => ( distinct_int(Xs)
       => ( sorted_wrt_int(ord_less_eq_int,Ys)
         => ( distinct_int(Ys)
           => ( ( aa_list_int_set_int(set_int2,Xs) = aa_list_int_set_int(set_int2,Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_5866_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] : ( aa_nat_VEBT_VEBT(vEBT_VEBT_insert(vEBT_Leaf((A),(B))),Xa) = aa_nat_VEBT_VEBT(vEBT_vebt_insert(vEBT_Leaf((A),(B))),Xa) ) ).

% VEBT_internal.insert'.simps(1)
tff(fact_5867_vebt__insert__code_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(vEBT_Leaf((A),(B))),Xa) = $ite(
          Xa = zero_zero_nat,
          vEBT_Leaf($true,(B)),
          $ite(Xa = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ) ).

% vebt_insert_code(1)
tff(fact_5868_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_s7982070591426661849_VEBTi(Ys))
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),I2) ) )
         => ( take_VEBT_VEBTi(K,Xs) = take_VEBT_VEBTi(K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_5869_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_int,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_size_list_int(Ys))
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
         => ( take_int(K,Xs) = take_int(K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_5870_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_s6755466524823107622T_VEBT(Ys))
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I2) ) )
         => ( take_VEBT_VEBT(K,Xs) = take_VEBT_VEBT(K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_5871_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_real,Ys: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_size_list_real(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_size_list_real(Ys))
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Ys),I2) ) )
         => ( take_real(K,Xs) = take_real(K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_5872_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),size_size_list_o(Ys))
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => ( aa_nat_o(nth_o(Xs),I2)
              <=> aa_nat_o(nth_o(Ys),I2) ) )
         => ( take_o(K,Xs) = take_o(K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_5873_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_list_nat_nat(size_size_list_nat,Ys))
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
         => ( take_nat(K,Xs) = take_nat(K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_5874_finite__sorted__distinct__unique,axiom,
    ! [A4: set_real] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ? [X: list_real] :
          ( ( aa_lis3769085826154824284t_real(set_real2,X) = A4 )
          & aa_list_real_o(sorted_wrt_real(ord_less_eq_real),X)
          & aa_list_real_o(distinct_real,X)
          & ! [Y4: list_real] :
              ( ( ( aa_lis3769085826154824284t_real(set_real2,Y4) = A4 )
                & aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Y4)
                & aa_list_real_o(distinct_real,Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5875_finite__sorted__distinct__unique,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ? [X: list_o] :
          ( ( aa_list_o_set_o(set_o2,X) = A4 )
          & aa_list_o_o(sorted_wrt_o(ord_less_eq_o),X)
          & aa_list_o_o(distinct_o,X)
          & ! [Y4: list_o] :
              ( ( ( aa_list_o_set_o(set_o2,Y4) = A4 )
                & aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Y4)
                & aa_list_o_o(distinct_o,Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5876_finite__sorted__distinct__unique,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ? [X: list_Code_integer] :
          ( ( set_Code_integer2(X) = A4 )
          & sorted710888440204495920nteger(ord_le3102999989581377725nteger,X)
          & distin1543349897113766820nteger(X)
          & ! [Y4: list_Code_integer] :
              ( ( ( set_Code_integer2(Y4) = A4 )
                & sorted710888440204495920nteger(ord_le3102999989581377725nteger,Y4)
                & distin1543349897113766820nteger(Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5877_finite__sorted__distinct__unique,axiom,
    ! [A4: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A4)
     => ? [X: list_rat] :
          ( ( set_rat2(X) = A4 )
          & sorted_wrt_rat(ord_less_eq_rat,X)
          & distinct_rat(X)
          & ! [Y4: list_rat] :
              ( ( ( set_rat2(Y4) = A4 )
                & sorted_wrt_rat(ord_less_eq_rat,Y4)
                & distinct_rat(Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5878_finite__sorted__distinct__unique,axiom,
    ! [A4: set_num] :
      ( aa_set_num_o(finite_finite_num,A4)
     => ? [X: list_num] :
          ( ( set_num2(X) = A4 )
          & sorted_wrt_num(ord_less_eq_num,X)
          & distinct_num(X)
          & ! [Y4: list_num] :
              ( ( ( set_num2(Y4) = A4 )
                & sorted_wrt_num(ord_less_eq_num,Y4)
                & distinct_num(Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5879_finite__sorted__distinct__unique,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ? [X: list_nat] :
          ( ( aa_list_nat_set_nat(set_nat2,X) = A4 )
          & aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),X)
          & aa_list_nat_o(distinct_nat,X)
          & ! [Y4: list_nat] :
              ( ( ( aa_list_nat_set_nat(set_nat2,Y4) = A4 )
                & aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Y4)
                & aa_list_nat_o(distinct_nat,Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5880_finite__sorted__distinct__unique,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ? [X: list_int] :
          ( ( aa_list_int_set_int(set_int2,X) = A4 )
          & sorted_wrt_int(ord_less_eq_int,X)
          & distinct_int(X)
          & ! [Y4: list_int] :
              ( ( ( aa_list_int_set_int(set_int2,Y4) = A4 )
                & sorted_wrt_int(ord_less_eq_int,Y4)
                & distinct_int(Y4) )
             => ( Y4 = X ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_5881_distinct__Ex1,axiom,
    ! [Xs: list_VEBT_VEBTi,Xa: vEBT_VEBTi] :
      ( distinct_VEBT_VEBTi(Xs)
     => ( member_VEBT_VEBTi(Xa,set_VEBT_VEBTi2(Xs))
       => ? [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),size_s7982070591426661849_VEBTi(Xs))
            & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),X) = Xa )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_s7982070591426661849_VEBTi(Xs))
                  & ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Y4) = Xa ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_5882_distinct__Ex1,axiom,
    ! [Xs: list_int,Xa: int] :
      ( distinct_int(Xs)
     => ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
       => ? [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),size_size_list_int(Xs))
            & ( aa_nat_int(nth_int(Xs),X) = Xa )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_size_list_int(Xs))
                  & ( aa_nat_int(nth_int(Xs),Y4) = Xa ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_5883_distinct__Ex1,axiom,
    ! [Xs: list_set_nat,Xa: set_nat] :
      ( distinct_set_nat(Xs)
     => ( aa_set_set_nat_o(member_set_nat(Xa),set_set_nat2(Xs))
       => ? [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),size_s3254054031482475050et_nat(Xs))
            & ( aa_nat_set_nat(nth_set_nat(Xs),X) = Xa )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_s3254054031482475050et_nat(Xs))
                  & ( aa_nat_set_nat(nth_set_nat(Xs),Y4) = Xa ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_5884_distinct__Ex1,axiom,
    ! [Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
       => ? [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),size_s6755466524823107622T_VEBT(Xs))
            & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),X) = Xa )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_s6755466524823107622T_VEBT(Xs))
                  & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Y4) = Xa ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_5885_distinct__Ex1,axiom,
    ! [Xs: list_real,Xa: real] :
      ( aa_list_real_o(distinct_real,Xs)
     => ( aa_set_real_o(member_real(Xa),aa_lis3769085826154824284t_real(set_real2,Xs))
       => ? [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),size_size_list_real(Xs))
            & ( aa_nat_real(nth_real(Xs),X) = Xa )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_size_list_real(Xs))
                  & ( aa_nat_real(nth_real(Xs),Y4) = Xa ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_5886_distinct__Ex1,axiom,
    ! [Xs: list_o,Xa: $o] :
      ( aa_list_o_o(distinct_o,Xs)
     => ( aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,Xs))
       => ? [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),size_size_list_o(Xs))
            & ( aa_nat_o(nth_o(Xs),X)
            <=> (Xa) )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),size_size_list_o(Xs))
                  & ( aa_nat_o(nth_o(Xs),Y4)
                  <=> (Xa) ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_5887_distinct__Ex1,axiom,
    ! [Xs: list_nat,Xa: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs))
       => ? [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_list_nat_nat(size_size_list_nat,Xs))
            & ( aa_nat_nat(nth_nat(Xs),X) = Xa )
            & ! [Y4: nat] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),aa_list_nat_nat(size_size_list_nat,Xs))
                  & ( aa_nat_nat(nth_nat(Xs),Y4) = Xa ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_5888_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_real] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),Xs)
     => ( aa_list_real_o(distinct_real,Xs)
       => ( linord4252657396651189596t_real(aa_lis3769085826154824284t_real(set_real2,Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5889_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_o] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),Xs)
     => ( aa_list_o_o(distinct_o,Xs)
       => ( linord3142498349692569832_set_o(aa_list_o_set_o(set_o2,Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5890_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => ( distinct_rat(Xs)
       => ( linord1979837681955606664et_rat(set_rat2(Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5891_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => ( distinct_num(Xs)
       => ( linord8395671565052656842et_num(set_num2(Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5892_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_int] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
     => ( distinct_int(Xs)
       => ( linord2612477271533052124et_int(aa_list_int_set_int(set_int2,Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5893_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
     => ( aa_list_nat_o(distinct_nat,Xs)
       => ( linord2614967742042102400et_nat(aa_list_nat_set_nat(set_nat2,Xs)) = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_5894_distinct__sorted__mono,axiom,
    ! [L: list_o,I: nat,J: nat] :
      ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),L)
     => ( aa_list_o_o(distinct_o,L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(L))
           => aa_o_o(aa_o_fun_o_o(ord_less_o,aa_nat_o(nth_o(L),I)),aa_nat_o(nth_o(L),J)) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_5895_distinct__sorted__mono,axiom,
    ! [L: list_real,I: nat,J: nat] :
      ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),L)
     => ( aa_list_real_o(distinct_real,L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(L))
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(nth_real(L),I)),aa_nat_real(nth_real(L),J)) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_5896_distinct__sorted__mono,axiom,
    ! [L: list_rat,I: nat,J: nat] :
      ( sorted_wrt_rat(ord_less_eq_rat,L)
     => ( distinct_rat(L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_rat(L))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(nth_rat(L),I)),aa_nat_rat(nth_rat(L),J)) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_5897_distinct__sorted__mono,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( sorted_wrt_num(ord_less_eq_num,L)
     => ( distinct_num(L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_num(L))
           => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(nth_num(L),I)),aa_nat_num(nth_num(L),J)) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_5898_distinct__sorted__mono,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),L)
     => ( aa_list_nat_o(distinct_nat,L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(nth_nat(L),I)),aa_nat_nat(nth_nat(L),J)) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_5899_distinct__sorted__mono,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( sorted_wrt_int(ord_less_eq_int,L)
     => ( distinct_int(L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(L))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(nth_int(L),I)),aa_nat_int(nth_int(L),J)) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_5900_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_o,I: nat,J: nat] :
      ( aa_list_o_o(distinct_o,L)
     => ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(L))
           => ( aa_o_o(aa_o_fun_o_o(ord_less_o,aa_nat_o(nth_o(L),I)),aa_nat_o(nth_o(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_5901_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_real,I: nat,J: nat] :
      ( aa_list_real_o(distinct_real,L)
     => ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(L))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(nth_real(L),I)),aa_nat_real(nth_real(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_5902_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_rat,I: nat,J: nat] :
      ( distinct_rat(L)
     => ( sorted_wrt_rat(ord_less_eq_rat,L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_rat(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_rat(L))
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(nth_rat(L),I)),aa_nat_rat(nth_rat(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_5903_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( distinct_num(L)
     => ( sorted_wrt_num(ord_less_eq_num,L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_num(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_num(L))
           => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(nth_num(L),I)),aa_nat_num(nth_num(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_5904_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( aa_list_nat_o(distinct_nat,L)
     => ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L))
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(nth_nat(L),I)),aa_nat_nat(nth_nat(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_5905_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( distinct_int(L)
     => ( sorted_wrt_int(ord_less_eq_int,L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(L))
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(nth_int(L),I)),aa_nat_int(nth_int(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_5906_distinct__sorted__mono__iff,axiom,
    ! [L: list_real,I: nat,J: nat] :
      ( aa_list_real_o(distinct_real,L)
     => ( aa_list_real_o(sorted_wrt_real(ord_less_eq_real),L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(L))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(nth_real(L),I)),aa_nat_real(nth_real(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_5907_distinct__sorted__mono__iff,axiom,
    ! [L: list_o,I: nat,J: nat] :
      ( aa_list_o_o(distinct_o,L)
     => ( aa_list_o_o(sorted_wrt_o(ord_less_eq_o),L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(L))
           => ( aa_o_o(aa_o_fun_o_o(ord_less_eq_o,aa_nat_o(nth_o(L),I)),aa_nat_o(nth_o(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_5908_distinct__sorted__mono__iff,axiom,
    ! [L: list_rat,I: nat,J: nat] :
      ( distinct_rat(L)
     => ( sorted_wrt_rat(ord_less_eq_rat,L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_rat(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_rat(L))
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(nth_rat(L),I)),aa_nat_rat(nth_rat(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_5909_distinct__sorted__mono__iff,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( distinct_num(L)
     => ( sorted_wrt_num(ord_less_eq_num,L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_num(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_num(L))
           => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(nth_num(L),I)),aa_nat_num(nth_num(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_5910_distinct__sorted__mono__iff,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( aa_list_nat_o(distinct_nat,L)
     => ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_list_nat_nat(size_size_list_nat,L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_list_nat_nat(size_size_list_nat,L))
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(nth_nat(L),I)),aa_nat_nat(nth_nat(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_5911_distinct__sorted__mono__iff,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( distinct_int(L)
     => ( sorted_wrt_int(ord_less_eq_int,L)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(L))
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(nth_int(L),I)),aa_nat_int(nth_int(L),J))
            <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_5912_distinct__list__update,axiom,
    ! [Xs: list_set_nat,A: set_nat,I: nat] :
      ( distinct_set_nat(Xs)
     => ( ~ aa_set_set_nat_o(member_set_nat(A),minus_2163939370556025621et_nat(set_set_nat2(Xs),aa_set1013257548857649334et_nat(insert_set_nat(aa_nat_set_nat(nth_set_nat(Xs),I)),bot_bot_set_set_nat)))
       => distinct_set_nat(list_update_set_nat(Xs,I,A)) ) ) ).

% distinct_list_update
tff(fact_5913_distinct__list__update,axiom,
    ! [Xs: list_real,A: real,I: nat] :
      ( aa_list_real_o(distinct_real,Xs)
     => ( ~ aa_set_real_o(member_real(A),minus_minus_set_real(aa_lis3769085826154824284t_real(set_real2,Xs),aa_set_real_set_real(insert_real(aa_nat_real(nth_real(Xs),I)),bot_bot_set_real)))
       => aa_list_real_o(distinct_real,list_update_real(Xs,I,A)) ) ) ).

% distinct_list_update
tff(fact_5914_distinct__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,A: vEBT_VEBTi,I: nat] :
      ( distinct_VEBT_VEBTi(Xs)
     => ( ~ member_VEBT_VEBTi(A,minus_3697805406911847364_VEBTi(set_VEBT_VEBTi2(Xs),insert_VEBT_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I),bot_bo8982466882572371071_VEBTi)))
       => distinct_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,A)) ) ) ).

% distinct_list_update
tff(fact_5915_distinct__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,A: vEBT_VEBT,I: nat] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),bot_bo8194388402131092736T_VEBT)))
       => aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,list_u1324408373059187874T_VEBT(Xs,I,A)) ) ) ).

% distinct_list_update
tff(fact_5916_distinct__list__update,axiom,
    ! [Xs: list_int,A: int,I: nat] :
      ( distinct_int(Xs)
     => ( ~ aa_set_int_o(member_int(A),minus_minus_set_int(aa_list_int_set_int(set_int2,Xs),aa_set_int_set_int(insert_int(aa_nat_int(nth_int(Xs),I)),bot_bot_set_int)))
       => distinct_int(list_update_int(Xs,I,A)) ) ) ).

% distinct_list_update
tff(fact_5917_distinct__list__update,axiom,
    ! [Xs: list_o,A: $o,I: nat] :
      ( aa_list_o_o(distinct_o,Xs)
     => ( ~ aa_set_o_o(member_o((A)),minus_minus_set_o(aa_list_o_set_o(set_o2,Xs),aa_set_o_set_o(insert_o(aa_nat_o(nth_o(Xs),I)),bot_bot_set_o)))
       => aa_list_o_o(distinct_o,list_update_o(Xs,I,(A))) ) ) ).

% distinct_list_update
tff(fact_5918_distinct__list__update,axiom,
    ! [Xs: list_nat,A: nat,I: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( ~ aa_set_nat_o(member_nat(A),minus_minus_set_nat(aa_list_nat_set_nat(set_nat2,Xs),aa_set_nat_set_nat(insert_nat(aa_nat_nat(nth_nat(Xs),I)),bot_bot_set_nat)))
       => aa_list_nat_o(distinct_nat,list_update_nat(Xs,I,A)) ) ) ).

% distinct_list_update
tff(fact_5919_vebt__insert_Osimps_I4_J,axiom,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] : ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary) ) ).

% vebt_insert.simps(4)
tff(fact_5920_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,Na: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(Xa,Tree_is)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xb))),
                  aa_o_assn(pure_assn,
                    ( ( none_P5556105721700978146at_nat = none_P5556105721700978146at_nat )
                    & ( Na = Na ) )))),aa_o_assn(pure_assn,Xc = vEBT_Nodei(none_P5556105721700978146at_nat,Na,Xa,Xb)))),H2)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Na,TreeList),Summary)),Xc)),H2) ) ).

% heaphelp
tff(fact_5921_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,Na: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(Xa,Tree_is)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Xb))),
                  aa_o_assn(pure_assn,
                    ( ( none_nat = none_nat )
                    & ( Na = Na ) )))),aa_o_assn(pure_assn,Xc = vEBT_Nodei(none_P5556105721700978146at_nat,Na,Xa,Xb)))),H2)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Na,TreeList),Summary)),Xc)),H2) ) ).

% heaphelp
tff(fact_5922_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,Sa: vEBT_VEBT,Xa: nat] : ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),Sa)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),Sa) ) ).

% vebt_insert.simps(3)
tff(fact_5923_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,Sa: vEBT_VEBT,Xa: nat] : ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),Sa)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),Sa) ) ).

% vebt_insert.simps(2)
tff(fact_5924_mergesort__remdups__correct,axiom,
    ! [L: list_real] :
      ( aa_list_real_o(distinct_real,merges7559785487730971421s_real(L))
      & aa_list_real_o(sorted_wrt_real(ord_less_eq_real),merges7559785487730971421s_real(L))
      & ( aa_lis3769085826154824284t_real(set_real2,merges7559785487730971421s_real(L)) = aa_lis3769085826154824284t_real(set_real2,L) ) ) ).

% mergesort_remdups_correct
tff(fact_5925_mergesort__remdups__correct,axiom,
    ! [L: list_o] :
      ( aa_list_o_o(distinct_o,mergesort_remdups_o(L))
      & aa_list_o_o(sorted_wrt_o(ord_less_eq_o),mergesort_remdups_o(L))
      & ( aa_list_o_set_o(set_o2,mergesort_remdups_o(L)) = aa_list_o_set_o(set_o2,L) ) ) ).

% mergesort_remdups_correct
tff(fact_5926_mergesort__remdups__correct,axiom,
    ! [L: list_rat] :
      ( distinct_rat(merges1021483306759835337ps_rat(L))
      & sorted_wrt_rat(ord_less_eq_rat,merges1021483306759835337ps_rat(L))
      & ( set_rat2(merges1021483306759835337ps_rat(L)) = set_rat2(L) ) ) ).

% mergesort_remdups_correct
tff(fact_5927_mergesort__remdups__correct,axiom,
    ! [L: list_num] :
      ( distinct_num(merges7437317189856885515ps_num(L))
      & sorted_wrt_num(ord_less_eq_num,merges7437317189856885515ps_num(L))
      & ( set_num2(merges7437317189856885515ps_num(L)) = set_num2(L) ) ) ).

% mergesort_remdups_correct
tff(fact_5928_mergesort__remdups__correct,axiom,
    ! [L: list_nat] :
      ( aa_list_nat_o(distinct_nat,merges1656613366846331073ps_nat(L))
      & aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),merges1656613366846331073ps_nat(L))
      & ( aa_list_nat_set_nat(set_nat2,merges1656613366846331073ps_nat(L)) = aa_list_nat_set_nat(set_nat2,L) ) ) ).

% mergesort_remdups_correct
tff(fact_5929_mergesort__remdups__correct,axiom,
    ! [L: list_int] :
      ( distinct_int(merges1654122896337280797ps_int(L))
      & sorted_wrt_int(ord_less_eq_int,merges1654122896337280797ps_int(L))
      & ( aa_list_int_set_int(set_int2,merges1654122896337280797ps_int(L)) = aa_list_int_set_int(set_int2,L) ) ) ).

% mergesort_remdups_correct
tff(fact_5930_last__take__nth__conv,axiom,
    ! [Na: nat,L: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s7982070591426661849_VEBTi(L))
     => ( ( Na != zero_zero_nat )
       => ( last_VEBT_VEBTi(take_VEBT_VEBTi(Na,L)) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_5931_last__take__nth__conv,axiom,
    ! [Na: nat,L: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_int(L))
     => ( ( Na != zero_zero_nat )
       => ( last_int(take_int(Na,L)) = aa_nat_int(nth_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_5932_last__take__nth__conv,axiom,
    ! [Na: nat,L: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s6755466524823107622T_VEBT(L))
     => ( ( Na != zero_zero_nat )
       => ( last_VEBT_VEBT(take_VEBT_VEBT(Na,L)) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_5933_last__take__nth__conv,axiom,
    ! [Na: nat,L: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_real(L))
     => ( ( Na != zero_zero_nat )
       => ( last_real(take_real(Na,L)) = aa_nat_real(nth_real(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_5934_last__take__nth__conv,axiom,
    ! [Na: nat,L: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_o(L))
     => ( ( Na != zero_zero_nat )
       => ( aa_list_o_o(last_o,take_o(Na,L))
        <=> aa_nat_o(nth_o(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_5935_last__take__nth__conv,axiom,
    ! [Na: nat,L: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_list_nat_nat(size_size_list_nat,L))
     => ( ( Na != zero_zero_nat )
       => ( aa_list_nat_nat(last_nat,take_nat(Na,L)) = aa_nat_nat(nth_nat(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ) ).

% last_take_nth_conv
tff(fact_5936_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( vEBT_Nodei(X11,X12,X13,X14) = vEBT_Nodei(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
tff(fact_5937_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd2: $o,Ve2: $o,V2: option4927543243414619207at_nat,Va2: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_Leaf((Vd2),(Ve2))),vEBT_Nodei(V2,Va2,Vb,Vc)) = bot_bot_assn ) ).

% vebt_assn_raw.simps(4)
tff(fact_5938_subset__subseqs,axiom,
    ! [X4: set_VEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(X4),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
     => member_set_VEBT_VEBT(X4,image_6463372868993444447T_VEBT(set_VEBT_VEBT2,set_list_VEBT_VEBT2(subseqs_VEBT_VEBT(Xs)))) ) ).

% subset_subseqs
tff(fact_5939_subset__subseqs,axiom,
    ! [X4: set_nat,Xs: list_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X4),aa_list_nat_set_nat(set_nat2,Xs))
     => aa_set_set_nat_o(member_set_nat(X4),image_1775855109352712557et_nat(set_nat2,set_list_nat2(subseqs_nat(Xs)))) ) ).

% subset_subseqs
tff(fact_5940_subset__subseqs,axiom,
    ! [X4: set_real,Xs: list_real] :
      ( aa_set_real_o(ord_less_eq_set_real(X4),aa_lis3769085826154824284t_real(set_real2,Xs))
     => member_set_real(X4,image_6239767680843085477t_real(set_real2,set_list_real2(subseqs_real(Xs)))) ) ).

% subset_subseqs
tff(fact_5941_subset__subseqs,axiom,
    ! [X4: set_o,Xs: list_o] :
      ( aa_set_o_o(ord_less_eq_set_o(X4),aa_list_o_set_o(set_o2,Xs))
     => member_set_o(X4,image_list_o_set_o(set_o2,set_list_o2(subseqs_o(Xs)))) ) ).

% subset_subseqs
tff(fact_5942_subset__subseqs,axiom,
    ! [X4: set_int,Xs: list_int] :
      ( aa_set_int_o(ord_less_eq_set_int(X4),aa_list_int_set_int(set_int2,Xs))
     => member_set_int(X4,image_3606813740839090725et_int(set_int2,set_list_int2(subseqs_int(Xs)))) ) ).

% subset_subseqs
tff(fact_5943_set__remove1__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Xs)
     => ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,remove1_VEBT_VEBT(Xa,Xs)) = minus_5127226145743854075T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT)) ) ) ).

% set_remove1_eq
tff(fact_5944_set__remove1__eq,axiom,
    ! [Xs: list_real,Xa: real] :
      ( aa_list_real_o(distinct_real,Xs)
     => ( aa_lis3769085826154824284t_real(set_real2,remove1_real(Xa,Xs)) = minus_minus_set_real(aa_lis3769085826154824284t_real(set_real2,Xs),aa_set_real_set_real(insert_real(Xa),bot_bot_set_real)) ) ) ).

% set_remove1_eq
tff(fact_5945_set__remove1__eq,axiom,
    ! [Xs: list_int,Xa: int] :
      ( distinct_int(Xs)
     => ( aa_list_int_set_int(set_int2,remove1_int(Xa,Xs)) = minus_minus_set_int(aa_list_int_set_int(set_int2,Xs),aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)) ) ) ).

% set_remove1_eq
tff(fact_5946_set__remove1__eq,axiom,
    ! [Xs: list_o,Xa: $o] :
      ( aa_list_o_o(distinct_o,Xs)
     => ( aa_list_o_set_o(set_o2,remove1_o((Xa),Xs)) = minus_minus_set_o(aa_list_o_set_o(set_o2,Xs),aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)) ) ) ).

% set_remove1_eq
tff(fact_5947_set__remove1__eq,axiom,
    ! [Xs: list_nat,Xa: nat] :
      ( aa_list_nat_o(distinct_nat,Xs)
     => ( aa_list_nat_set_nat(set_nat2,remove1_nat(Xa,Xs)) = minus_minus_set_nat(aa_list_nat_set_nat(set_nat2,Xs),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) ) ) ).

% set_remove1_eq
tff(fact_5948_lex__take__index,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi,R: set_Pr2227491710730465451_VEBTi] :
      ( member4173000155140927252_VEBTi(produc4384243565435462691_VEBTi(Xs,Ys),lex_VEBT_VEBTi(R))
     => ~ ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Ys))
             => ( ( take_VEBT_VEBTi(I2,Xs) = take_VEBT_VEBTi(I2,Ys) )
               => ~ member660371905731732212_VEBTi(produc436343169921013763_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I2),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),I2)),R) ) ) ) ) ).

% lex_take_index
tff(fact_5949_lex__take__index,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( member2333554998283850498uint32(produc7487160679990061969uint32(Xs,Ys),lex_uint32(R))
     => ~ ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s4844771616002835472uint32(Xs))
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s4844771616002835472uint32(Ys))
             => ( ( take_uint32(I2,Xs) = take_uint32(I2,Ys) )
               => ~ member8027108493173000802uint32(produc1400373151660368625uint32(aa_nat_uint32(nth_uint32(Xs),I2),aa_nat_uint32(nth_uint32(Ys),I2)),R) ) ) ) ) ).

% lex_take_index
tff(fact_5950_lex__take__index,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),lex_int(R))
     => ~ ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Ys))
             => ( ( take_int(I2,Xs) = take_int(I2,Ys) )
               => ~ member5262025264175285858nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),I2),aa_nat_int(nth_int(Ys),I2)),R) ) ) ) ) ).

% lex_take_index
tff(fact_5951_lex__take__index,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),lex_VEBT_VEBT(R))
     => ~ ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Ys))
             => ( ( take_VEBT_VEBT(I2,Xs) = take_VEBT_VEBT(I2,Ys) )
               => ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I2)),R) ) ) ) ) ).

% lex_take_index
tff(fact_5952_lex__take__index,axiom,
    ! [Xs: list_real,Ys: list_real,R: set_Pr6218003697084177305l_real] :
      ( member6584958104391596930t_real(produc1408950526243324945t_real(Xs,Ys),lex_real(R))
     => ~ ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Ys))
             => ( ( take_real(I2,Xs) = take_real(I2,Ys) )
               => ~ member7849222048561428706l_real(produc4511245868158468465l_real(aa_nat_real(nth_real(Xs),I2),aa_nat_real(nth_real(Ys),I2)),R) ) ) ) ) ).

% lex_take_index
tff(fact_5953_lex__take__index,axiom,
    ! [Xs: list_o,Ys: list_o,R: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(Xs,Ys),lex_o(R))
     => ~ ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Ys))
             => ( ( take_o(I2,Xs) = take_o(I2,Ys) )
               => ~ member7466972457876170832od_o_o(product_Pair_o_o(aa_nat_o(nth_o(Xs),I2),aa_nat_o(nth_o(Ys),I2)),R) ) ) ) ) ).

% lex_take_index
tff(fact_5954_lex__take__index,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),lex_nat(R))
     => ~ ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Ys))
             => ( ( take_nat(I2,Xs) = take_nat(I2,Ys) )
               => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Ys),I2)),R) ) ) ) ) ).

% lex_take_index
tff(fact_5955_take__butlast__conv,axiom,
    ! [L: list_VEBT_VEBT] : ( take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s6755466524823107622T_VEBT(L)),aa_nat_nat(suc,zero_zero_nat)),L) = butlast_VEBT_VEBT(L) ) ).

% take_butlast_conv
tff(fact_5956_take__butlast__conv,axiom,
    ! [L: list_real] : ( take_real(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_real(L)),aa_nat_nat(suc,zero_zero_nat)),L) = butlast_real(L) ) ).

% take_butlast_conv
tff(fact_5957_take__butlast__conv,axiom,
    ! [L: list_o] : ( take_o(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_o(L)),aa_nat_nat(suc,zero_zero_nat)),L) = butlast_o(L) ) ).

% take_butlast_conv
tff(fact_5958_take__butlast__conv,axiom,
    ! [L: list_nat] : ( take_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_nat_nat(size_size_list_nat,L)),aa_nat_nat(suc,zero_zero_nat)),L) = butlast_nat(L) ) ).

% take_butlast_conv
tff(fact_5959_take__minus__one__conv__butlast,axiom,
    ! [Na: nat,L: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s6755466524823107622T_VEBT(L))
     => ( take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)),L) = butlast_VEBT_VEBT(take_VEBT_VEBT(Na,L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_5960_take__minus__one__conv__butlast,axiom,
    ! [Na: nat,L: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_real(L))
     => ( take_real(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)),L) = butlast_real(take_real(Na,L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_5961_take__minus__one__conv__butlast,axiom,
    ! [Na: nat,L: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_o(L))
     => ( take_o(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)),L) = butlast_o(take_o(Na,L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_5962_take__minus__one__conv__butlast,axiom,
    ! [Na: nat,L: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_list_nat_nat(size_size_list_nat,L))
     => ( take_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)),L) = butlast_nat(take_nat(Na,L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_5963_Cons__in__lex,axiom,
    ! [Xa: uint32,Xs: list_uint32,Y: uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( member2333554998283850498uint32(produc7487160679990061969uint32(cons_uint32(Xa,Xs),cons_uint32(Y,Ys)),lex_uint32(R))
    <=> ( ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y),R)
          & ( size_s4844771616002835472uint32(Xs) = size_s4844771616002835472uint32(Ys) ) )
        | ( ( Xa = Y )
          & member2333554998283850498uint32(produc7487160679990061969uint32(Xs,Ys),lex_uint32(R)) ) ) ) ).

% Cons_in_lex
tff(fact_5964_Cons__in__lex,axiom,
    ! [Xa: int,Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(Xa,Xs),cons_int(Y,Ys)),lex_int(R))
    <=> ( ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Y),R)
          & ( size_size_list_int(Xs) = size_size_list_int(Ys) ) )
        | ( ( Xa = Y )
          & member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),lex_int(R)) ) ) ) ).

% Cons_in_lex
tff(fact_5965_Cons__in__lex,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT,Ys: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(cons_VEBT_VEBT(Xa,Xs),cons_VEBT_VEBT(Y,Ys)),lex_VEBT_VEBT(R))
    <=> ( ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(Xa,Y),R)
          & ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) )
        | ( ( Xa = Y )
          & member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),lex_VEBT_VEBT(R)) ) ) ) ).

% Cons_in_lex
tff(fact_5966_Cons__in__lex,axiom,
    ! [Xa: real,Xs: list_real,Y: real,Ys: list_real,R: set_Pr6218003697084177305l_real] :
      ( member6584958104391596930t_real(produc1408950526243324945t_real(cons_real(Xa,Xs),cons_real(Y,Ys)),lex_real(R))
    <=> ( ( member7849222048561428706l_real(produc4511245868158468465l_real(Xa,Y),R)
          & ( size_size_list_real(Xs) = size_size_list_real(Ys) ) )
        | ( ( Xa = Y )
          & member6584958104391596930t_real(produc1408950526243324945t_real(Xs,Ys),lex_real(R)) ) ) ) ).

% Cons_in_lex
tff(fact_5967_Cons__in__lex,axiom,
    ! [Xa: $o,Xs: list_o,Y: $o,Ys: list_o,R: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(cons_o((Xa),Xs),cons_o((Y),Ys)),lex_o(R))
    <=> ( ( member7466972457876170832od_o_o(product_Pair_o_o((Xa),(Y)),R)
          & ( size_size_list_o(Xs) = size_size_list_o(Ys) ) )
        | ( ( (Xa)
          <=> (Y) )
          & member4159035015898711888list_o(produc8435520187683070743list_o(Xs,Ys),lex_o(R)) ) ) ) ).

% Cons_in_lex
tff(fact_5968_Cons__in__lex,axiom,
    ! [Xa: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(Xa,Xs),cons_nat(Y,Ys)),lex_nat(R))
    <=> ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y),R)
          & ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) ) )
        | ( ( Xa = Y )
          & member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),lex_nat(R)) ) ) ) ).

% Cons_in_lex
tff(fact_5969_butlast__update_H,axiom,
    ! [L: list_nat,I: nat,Xa: nat] : ( list_update_nat(butlast_nat(L),I,Xa) = butlast_nat(list_update_nat(L,I,Xa)) ) ).

% butlast_update'
tff(fact_5970_butlast__update_H,axiom,
    ! [L: list_VEBT_VEBTi,I: nat,Xa: vEBT_VEBTi] : ( list_u6098035379799741383_VEBTi(butlast_VEBT_VEBTi(L),I,Xa) = butlast_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,Xa)) ) ).

% butlast_update'
tff(fact_5971_butlast__update_H,axiom,
    ! [L: list_VEBT_VEBT,I: nat,Xa: vEBT_VEBT] : ( list_u1324408373059187874T_VEBT(butlast_VEBT_VEBT(L),I,Xa) = butlast_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,Xa)) ) ).

% butlast_update'
tff(fact_5972_foldl__absorb1,axiom,
    ! [Xa: rat,Zs: list_rat] : ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xa),foldl_rat_rat(times_times_rat,one_one_rat,Zs)) = foldl_rat_rat(times_times_rat,Xa,Zs) ) ).

% foldl_absorb1
tff(fact_5973_foldl__absorb1,axiom,
    ! [Xa: assn,Zs: list_assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Xa),foldl_assn_assn(times_times_assn,one_one_assn,Zs)) = foldl_assn_assn(times_times_assn,Xa,Zs) ) ).

% foldl_absorb1
tff(fact_5974_foldl__absorb1,axiom,
    ! [Xa: uint32,Zs: list_uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,Xa),foldl_uint32_uint32(times_times_uint32,one_one_uint32,Zs)) = foldl_uint32_uint32(times_times_uint32,Xa,Zs) ) ).

% foldl_absorb1
tff(fact_5975_foldl__absorb1,axiom,
    ! [Xa: real,Zs: list_real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),foldl_real_real(times_times_real,one_one_real,Zs)) = foldl_real_real(times_times_real,Xa,Zs) ) ).

% foldl_absorb1
tff(fact_5976_foldl__absorb1,axiom,
    ! [Xa: nat,Zs: list_nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),aa_list_nat_nat(foldl_nat_nat(times_times_nat,one_one_nat),Zs)) = aa_list_nat_nat(foldl_nat_nat(times_times_nat,Xa),Zs) ) ).

% foldl_absorb1
tff(fact_5977_foldl__absorb1,axiom,
    ! [Xa: int,Zs: list_int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,Xa),foldl_int_int(times_times_int,one_one_int,Zs)) = foldl_int_int(times_times_int,Xa,Zs) ) ).

% foldl_absorb1
tff(fact_5978_set__remove1__subset,axiom,
    ! [Xa: vEBT_VEBT,Xs: list_VEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,remove1_VEBT_VEBT(Xa,Xs))),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs)) ).

% set_remove1_subset
tff(fact_5979_set__remove1__subset,axiom,
    ! [Xa: nat,Xs: list_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,remove1_nat(Xa,Xs))),aa_list_nat_set_nat(set_nat2,Xs)) ).

% set_remove1_subset
tff(fact_5980_set__remove1__subset,axiom,
    ! [Xa: real,Xs: list_real] : aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,remove1_real(Xa,Xs))),aa_lis3769085826154824284t_real(set_real2,Xs)) ).

% set_remove1_subset
tff(fact_5981_set__remove1__subset,axiom,
    ! [Xa: $o,Xs: list_o] : aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,remove1_o((Xa),Xs))),aa_list_o_set_o(set_o2,Xs)) ).

% set_remove1_subset
tff(fact_5982_set__remove1__subset,axiom,
    ! [Xa: int,Xs: list_int] : aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,remove1_int(Xa,Xs))),aa_list_int_set_int(set_int2,Xs)) ).

% set_remove1_subset
tff(fact_5983_sorted__remove1,axiom,
    ! [Xs: list_rat,A: rat] :
      ( sorted_wrt_rat(ord_less_eq_rat,Xs)
     => sorted_wrt_rat(ord_less_eq_rat,remove1_rat(A,Xs)) ) ).

% sorted_remove1
tff(fact_5984_sorted__remove1,axiom,
    ! [Xs: list_num,A: num] :
      ( sorted_wrt_num(ord_less_eq_num,Xs)
     => sorted_wrt_num(ord_less_eq_num,remove1_num(A,Xs)) ) ).

% sorted_remove1
tff(fact_5985_sorted__remove1,axiom,
    ! [Xs: list_nat,A: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),Xs)
     => aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),remove1_nat(A,Xs)) ) ).

% sorted_remove1
tff(fact_5986_sorted__remove1,axiom,
    ! [Xs: list_int,A: int] :
      ( sorted_wrt_int(ord_less_eq_int,Xs)
     => sorted_wrt_int(ord_less_eq_int,remove1_int(A,Xs)) ) ).

% sorted_remove1
tff(fact_5987_nth__butlast,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s7982070591426661849_VEBTi(butlast_VEBT_VEBTi(Xs)))
     => ( aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(butlast_VEBT_VEBTi(Xs)),Na) = aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),Na) ) ) ).

% nth_butlast
tff(fact_5988_nth__butlast,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(butlast_int(Xs)))
     => ( aa_nat_int(nth_int(butlast_int(Xs)),Na) = aa_nat_int(nth_int(Xs),Na) ) ) ).

% nth_butlast
tff(fact_5989_nth__butlast,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(butlast_VEBT_VEBT(Xs)))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(butlast_VEBT_VEBT(Xs)),Na) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Na) ) ) ).

% nth_butlast
tff(fact_5990_nth__butlast,axiom,
    ! [Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(butlast_real(Xs)))
     => ( aa_nat_real(nth_real(butlast_real(Xs)),Na) = aa_nat_real(nth_real(Xs),Na) ) ) ).

% nth_butlast
tff(fact_5991_nth__butlast,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(butlast_o(Xs)))
     => ( aa_nat_o(nth_o(butlast_o(Xs)),Na)
      <=> aa_nat_o(nth_o(Xs),Na) ) ) ).

% nth_butlast
tff(fact_5992_nth__butlast,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,butlast_nat(Xs)))
     => ( aa_nat_nat(nth_nat(butlast_nat(Xs)),Na) = aa_nat_nat(nth_nat(Xs),Na) ) ) ).

% nth_butlast
tff(fact_5993_take__butlast,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => ( take_VEBT_VEBT(Na,butlast_VEBT_VEBT(Xs)) = take_VEBT_VEBT(Na,Xs) ) ) ).

% take_butlast
tff(fact_5994_take__butlast,axiom,
    ! [Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => ( take_real(Na,butlast_real(Xs)) = take_real(Na,Xs) ) ) ).

% take_butlast
tff(fact_5995_take__butlast,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => ( take_o(Na,butlast_o(Xs)) = take_o(Na,Xs) ) ) ).

% take_butlast
tff(fact_5996_take__butlast,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( take_nat(Na,butlast_nat(Xs)) = take_nat(Na,Xs) ) ) ).

% take_butlast
tff(fact_5997_distinct__butlast__swap,axiom,
    ! [Pq: list_nat,I: nat] :
      ( aa_list_nat_o(distinct_nat,Pq)
     => aa_list_nat_o(distinct_nat,butlast_nat(list_update_nat(Pq,I,aa_list_nat_nat(last_nat,Pq)))) ) ).

% distinct_butlast_swap
tff(fact_5998_distinct__butlast__swap,axiom,
    ! [Pq: list_VEBT_VEBTi,I: nat] :
      ( distinct_VEBT_VEBTi(Pq)
     => distinct_VEBT_VEBTi(butlast_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Pq,I,last_VEBT_VEBTi(Pq)))) ) ).

% distinct_butlast_swap
tff(fact_5999_distinct__butlast__swap,axiom,
    ! [Pq: list_VEBT_VEBT,I: nat] :
      ( aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Pq)
     => aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,butlast_VEBT_VEBT(list_u1324408373059187874T_VEBT(Pq,I,last_VEBT_VEBT(Pq)))) ) ).

% distinct_butlast_swap
tff(fact_6000_butlast__take,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => ( butlast_VEBT_VEBT(take_VEBT_VEBT(Na,Xs)) = take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),Xs) ) ) ).

% butlast_take
tff(fact_6001_butlast__take,axiom,
    ! [Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_real(Xs))
     => ( butlast_real(take_real(Na,Xs)) = take_real(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),Xs) ) ) ).

% butlast_take
tff(fact_6002_butlast__take,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_o(Xs))
     => ( butlast_o(take_o(Na,Xs)) = take_o(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),Xs) ) ) ).

% butlast_take
tff(fact_6003_butlast__take,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_list_nat_nat(size_size_list_nat,Xs))
     => ( butlast_nat(take_nat(Na,Xs)) = take_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),Xs) ) ) ).

% butlast_take
tff(fact_6004_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A4: set_Code_integer,Xa: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( linord2324613341767563021nteger(minus_2355218937544613996nteger(A4,insert_Code_integer(Xa,bot_bo3990330152332043303nteger))) = remove1_Code_integer(Xa,linord2324613341767563021nteger(A4)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_6005_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A4: set_int,Xa: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( linord2612477271533052124et_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int))) = remove1_int(Xa,linord2612477271533052124et_int(A4)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_6006_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A4: set_o,Xa: $o] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( linord3142498349692569832_set_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o))) = remove1_o((Xa),linord3142498349692569832_set_o(A4)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_6007_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( linord2614967742042102400et_nat(minus_minus_set_nat(A4,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat))) = remove1_nat(Xa,linord2614967742042102400et_nat(A4)) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_6008_Cons__lenlex__iff,axiom,
    ! [M: uint32,Ms: list_uint32,Na: uint32,Ns: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( member2333554998283850498uint32(produc7487160679990061969uint32(cons_uint32(M,Ms),cons_uint32(Na,Ns)),lenlex_uint32(R))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s4844771616002835472uint32(Ms)),size_s4844771616002835472uint32(Ns))
        | ( ( size_s4844771616002835472uint32(Ms) = size_s4844771616002835472uint32(Ns) )
          & member8027108493173000802uint32(produc1400373151660368625uint32(M,Na),R) )
        | ( ( M = Na )
          & member2333554998283850498uint32(produc7487160679990061969uint32(Ms,Ns),lenlex_uint32(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6009_Cons__lenlex__iff,axiom,
    ! [M: int,Ms: list_int,Na: int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(M,Ms),cons_int(Na,Ns)),lenlex_int(R))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ms)),size_size_list_int(Ns))
        | ( ( size_size_list_int(Ms) = size_size_list_int(Ns) )
          & member5262025264175285858nt_int(product_Pair_int_int(M,Na),R) )
        | ( ( M = Na )
          & member6698963635872716290st_int(produc364263696895485585st_int(Ms,Ns),lenlex_int(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6010_Cons__lenlex__iff,axiom,
    ! [M: vEBT_VEBT,Ms: list_VEBT_VEBT,Na: vEBT_VEBT,Ns: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(cons_VEBT_VEBT(M,Ms),cons_VEBT_VEBT(Na,Ns)),lenlex_VEBT_VEBT(R))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ms)),size_s6755466524823107622T_VEBT(Ns))
        | ( ( size_s6755466524823107622T_VEBT(Ms) = size_s6755466524823107622T_VEBT(Ns) )
          & member568628332442017744T_VEBT(produc537772716801021591T_VEBT(M,Na),R) )
        | ( ( M = Na )
          & member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Ms,Ns),lenlex_VEBT_VEBT(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6011_Cons__lenlex__iff,axiom,
    ! [M: real,Ms: list_real,Na: real,Ns: list_real,R: set_Pr6218003697084177305l_real] :
      ( member6584958104391596930t_real(produc1408950526243324945t_real(cons_real(M,Ms),cons_real(Na,Ns)),lenlex_real(R))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(Ms)),size_size_list_real(Ns))
        | ( ( size_size_list_real(Ms) = size_size_list_real(Ns) )
          & member7849222048561428706l_real(produc4511245868158468465l_real(M,Na),R) )
        | ( ( M = Na )
          & member6584958104391596930t_real(produc1408950526243324945t_real(Ms,Ns),lenlex_real(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6012_Cons__lenlex__iff,axiom,
    ! [M: $o,Ms: list_o,Na: $o,Ns: list_o,R: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(cons_o((M),Ms),cons_o((Na),Ns)),lenlex_o(R))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ms)),size_size_list_o(Ns))
        | ( ( size_size_list_o(Ms) = size_size_list_o(Ns) )
          & member7466972457876170832od_o_o(product_Pair_o_o((M),(Na)),R) )
        | ( ( (M)
          <=> (Na) )
          & member4159035015898711888list_o(produc8435520187683070743list_o(Ms,Ns),lenlex_o(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6013_Cons__lenlex__iff,axiom,
    ! [M: nat,Ms: list_nat,Na: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(M,Ms),cons_nat(Na,Ns)),lenlex_nat(R))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_list_nat_nat(size_size_list_nat,Ms)),aa_list_nat_nat(size_size_list_nat,Ns))
        | ( ( aa_list_nat_nat(size_size_list_nat,Ms) = aa_list_nat_nat(size_size_list_nat,Ns) )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),Na),R) )
        | ( ( M = Na )
          & member7340969449405702474st_nat(produc2694037385005941721st_nat(Ms,Ns),lenlex_nat(R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6014_listrel1__iff__update,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi,R: set_Pr2227491710730465451_VEBTi] :
      ( member4173000155140927252_VEBTi(produc4384243565435462691_VEBTi(Xs,Ys),listrel1_VEBT_VEBTi(R))
    <=> ? [Y5: vEBT_VEBTi,N5: nat] :
          ( member660371905731732212_VEBTi(produc436343169921013763_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),N5),Y5),R)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),size_s7982070591426661849_VEBTi(Xs))
          & ( Ys = list_u6098035379799741383_VEBTi(Xs,N5,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_6015_listrel1__iff__update,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( member2333554998283850498uint32(produc7487160679990061969uint32(Xs,Ys),listrel1_uint32(R))
    <=> ? [Y5: uint32,N5: nat] :
          ( member8027108493173000802uint32(produc1400373151660368625uint32(aa_nat_uint32(nth_uint32(Xs),N5),Y5),R)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),size_s4844771616002835472uint32(Xs))
          & ( Ys = list_update_uint32(Xs,N5,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_6016_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R))
    <=> ? [Y5: int,N5: nat] :
          ( member5262025264175285858nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),N5),Y5),R)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),size_size_list_int(Xs))
          & ( Ys = list_update_int(Xs,N5,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_6017_listrel1__iff__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys),listrel1_VEBT_VEBT(R))
    <=> ? [Y5: vEBT_VEBT,N5: nat] :
          ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),N5),Y5),R)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),size_s6755466524823107622T_VEBT(Xs))
          & ( Ys = list_u1324408373059187874T_VEBT(Xs,N5,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_6018_listrel1__iff__update,axiom,
    ! [Xs: list_real,Ys: list_real,R: set_Pr6218003697084177305l_real] :
      ( member6584958104391596930t_real(produc1408950526243324945t_real(Xs,Ys),listrel1_real(R))
    <=> ? [Y5: real,N5: nat] :
          ( member7849222048561428706l_real(produc4511245868158468465l_real(aa_nat_real(nth_real(Xs),N5),Y5),R)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),size_size_list_real(Xs))
          & ( Ys = list_update_real(Xs,N5,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_6019_listrel1__iff__update,axiom,
    ! [Xs: list_o,Ys: list_o,R: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(Xs,Ys),listrel1_o(R))
    <=> ? [Y5: $o,N5: nat] :
          ( member7466972457876170832od_o_o(product_Pair_o_o(aa_nat_o(nth_o(Xs),N5),(Y5)),R)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),size_size_list_o(Xs))
          & ( Ys = list_update_o(Xs,N5,(Y5)) ) ) ) ).

% listrel1_iff_update
tff(fact_6020_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R))
    <=> ? [Y5: nat,N5: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),N5)),Y5),R)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),aa_list_nat_nat(size_size_list_nat,Xs))
          & ( Ys = list_update_nat(Xs,N5,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_6021_total__on__singleton,axiom,
    ! [Xa: vEBT_VEBT] : total_on_VEBT_VEBT(aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Xa),bot_bo8194388402131092736T_VEBT),insert494605675473494903T_VEBT(produc537772716801021591T_VEBT(Xa,Xa),bot_bo5088076668136028147T_VEBT)) ).

% total_on_singleton
tff(fact_6022_total__on__singleton,axiom,
    ! [Xa: uint32] : total_on_uint32(insert_uint32(Xa,bot_bot_set_uint32),insert4454361187789264009uint32(produc1400373151660368625uint32(Xa,Xa),bot_bo8438649754162204037uint32)) ).

% total_on_singleton
tff(fact_6023_total__on__singleton,axiom,
    ! [Xa: nat] : total_on_nat(aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat),insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa),bot_bo2099793752762293965at_nat)) ).

% total_on_singleton
tff(fact_6024_total__on__singleton,axiom,
    ! [Xa: int] : total_on_int(aa_set_int_set_int(insert_int(Xa),bot_bot_set_int),insert5033312907999012233nt_int(product_Pair_int_int(Xa,Xa),bot_bo1796632182523588997nt_int)) ).

% total_on_singleton
tff(fact_6025_total__on__singleton,axiom,
    ! [Xa: $o] : total_on_o(aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o),insert6201435330877294327od_o_o(product_Pair_o_o((Xa),(Xa)),bot_bo7073875226086086771od_o_o)) ).

% total_on_singleton
tff(fact_6026_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Ys))
       => ( nth_Pr6329974346453275474_VEBTi(zip_VE793581609497812771_VEBTi(Xs,Ys),I) = produc436343169921013763_VEBTi(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6027_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Ys))
       => ( nth_Pr2733638074192486473Ti_int(zip_VEBT_VEBTi_int(Xs,Ys),I) = produc7190175284220460154Ti_int(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I),aa_nat_int(nth_int(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6028_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Ys))
       => ( nth_Pr5092243222550848201_VEBTi(zip_int_VEBT_VEBTi(Xs,Ys),I) = produc4497823428076633978_VEBTi(aa_nat_int(nth_int(Xs),I),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6029_nth__zip,axiom,
    ! [I: nat,Xs: list_uint32,Ys: list_uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s4844771616002835472uint32(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s4844771616002835472uint32(Ys))
       => ( nth_Pr6139322253122866368uint32(zip_uint32_uint32(Xs,Ys),I) = produc1400373151660368625uint32(aa_nat_uint32(nth_uint32(Xs),I),aa_nat_uint32(nth_uint32(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6030_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Ys))
       => ( nth_Pr4439495888332055232nt_int(zip_int_int(Xs,Ys),I) = product_Pair_int_int(aa_nat_int(nth_int(Xs),I),aa_nat_int(nth_int(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6031_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ys))
       => ( nth_Pr8725177398587324397T_VEBT(zip_VE7413257051550508102T_VEBT(Xs,Ys),I) = produc7053807326796202854T_VEBT(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6032_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ys))
       => ( nth_Pr3474266648193625910T_VEBT(zip_int_VEBT_VEBT(Xs,Ys),I) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),I),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6033_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Ys))
       => ( nth_Pr3433448822664029129i_real(zip_VEBT_VEBTi_real(Xs,Ys),I) = produc8457151488442208762i_real(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I),aa_nat_real(nth_real(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6034_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Ys))
       => ( nth_Pr731366597535767232t_real(zip_int_real(Xs,Ys),I) = produc801115645435158769t_real(aa_nat_int(nth_int(Xs),I),aa_nat_real(nth_real(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6035_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Ys))
       => ( nth_Pr3306050735993963089EBTi_o(zip_VEBT_VEBTi_o(Xs,Ys),I) = produc8194178580519725514EBTi_o(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I),aa_nat_o(nth_o(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_6036_rule__at__index,axiom,
    ! [P2: assn,A4: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7265847600308530106T_VEBT(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis4124290615194246587i_assn(aTP_Lamm_ag(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6037_rule__at__index,axiom,
    ! [P2: assn,A4: fun_VE1225457145571040039t_assn,Xs: list_VEBT_VEBTi,Xsi: list_nat,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8930081998596925642Ti_nat(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8068455526268438248t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2809031099982602151Ti_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis9001725492116066833i_assn(aTP_Lamm_ah(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6038_rule__at__index,axiom,
    ! [P2: assn,A4: fun_VE2561185598974572172i_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1891944875198410415_VEBTi(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis6960512757483956916i_assn(aa_lis2738398937141596790i_assn(aTP_Lamm_ai(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6039_rule__at__index,axiom,
    ! [P2: assn,A4: fun_VE5641251438918192003t_assn,Xs: list_VEBT_VEBTi,Xsi: list_int,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8927591528087875366Ti_int(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_VEB9043990991474615620t_assn(A4,aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L2806540629473551875Ti_int(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis3326845930971794347i_assn(aa_lis2594938629241682541i_assn(aTP_Lamm_aj(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6040_rule__at__index,axiom,
    ! [P2: assn,A4: fun_in7755481285845205148T_assn,Xs: list_int,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1664421287176695555T_VEBT(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2018189785592951398T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_int(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L2018189785592951398T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_int(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis9004913286454700036i_assn(aTP_Lamm_ak(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6041_rule__at__index,axiom,
    ! [P2: assn,A4: fun_int_fun_nat_assn,Xs: list_int,Xsi: list_nat,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L77084186935402305nt_nat(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8891422820522952478nt_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_int(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8891422820522952478nt_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_int(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis5475328533724316040i_assn(aTP_Lamm_al(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6042_rule__at__index,axiom,
    ! [P2: assn,A4: fun_in1903134756389476099i_assn,Xs: list_int,Xsi: list_VEBT_VEBTi,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6235239671944049190_VEBTi(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L114188773329725699_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_int(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_int3579307910570193220i_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Xsi),I))),vEBT_L114188773329725699_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_int(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis6960512757483956916i_assn(aa_lis2664486819095662317i_assn(aTP_Lamm_am(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6043_rule__at__index,axiom,
    ! [P2: assn,A4: fun_int_fun_int_assn,Xs: list_int,Xsi: list_int,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L74593716426352029nt_int(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8888932350013902202nt_int(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_int(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(A4,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8888932350013902202nt_int(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_int(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis3326845930971794347i_assn(aa_lis8291913707704707556i_assn(aTP_Lamm_an(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6044_rule__at__index,axiom,
    ! [P2: assn,A4: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1279224858307276611T_VEBT(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis7580683478039494528i_assn(aTP_Lamm_ao(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6045_rule__at__index,axiom,
    ! [P2: assn,A4: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,Xsi: list_nat,F2: assn,I: nat,C2: heap_T8145700208782473153_VEBTi,Q4: fun_VEBT_VEBTi_assn,F4: fun_VEBT_VEBTi_assn] :
      ( entails(P2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8296926524756676353BT_nat(A4,Xs,Xsi)),F2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),F2),C2,Q4)
         => ( ! [R3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q4,R3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_VEB8640158415945897581t_assn(A4,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)),A4,Xs,Xsi))),aa_VEBT_VEBTi_assn(F4,R3)))
           => hoare_1429296392585015714_VEBTi(P2,C2,aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis3335532474104912652i_assn(aTP_Lamm_ap(A4),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_6046_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Na),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ) ).

% atLeast1_lessThan_eq_remove0
tff(fact_6047_Bernoulli__inequality,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),Na)) ) ).

% Bernoulli_inequality
tff(fact_6048_set__vebt_H__def,axiom,
    ! [T: vEBT_VEBT] : ( vEBT_VEBT_set_vebt(T) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T)) ) ).

% set_vebt'_def
tff(fact_6049_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_6050_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_6051_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_6052_image__ident,axiom,
    ! [Y6: set_nat] : ( aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_aq),Y6) = Y6 ) ).

% image_ident
tff(fact_6053_image__ident,axiom,
    ! [Y6: set_int] : ( aa_set_int_set_int(image_int_int(aTP_Lamm_ar),Y6) = Y6 ) ).

% image_ident
tff(fact_6054_finite__Collect__disjI,axiom,
    ! [P2: fun_list_nat_o,Q2: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_as(P2),Q2)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P2))
        & finite8100373058378681591st_nat(collect_list_nat(Q2)) ) ) ).

% finite_Collect_disjI
tff(fact_6055_finite__Collect__disjI,axiom,
    ! [P2: fun_set_nat_o,Q2: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_at(P2),Q2)))
    <=> ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P2))
        & aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q2)) ) ) ).

% finite_Collect_disjI
tff(fact_6056_finite__Collect__disjI,axiom,
    ! [P2: fun_nat_o,Q2: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_au(P2),Q2)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q2)) ) ) ).

% finite_Collect_disjI
tff(fact_6057_finite__Collect__disjI,axiom,
    ! [P2: fun_int_o,Q2: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_av(P2),Q2)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(P2))
        & aa_set_int_o(finite_finite_int,collect_int(Q2)) ) ) ).

% finite_Collect_disjI
tff(fact_6058_finite__Collect__disjI,axiom,
    ! [P2: fun_complex_o,Q2: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_aw(P2),Q2)))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P2))
        & aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q2)) ) ) ).

% finite_Collect_disjI
tff(fact_6059_finite__Collect__disjI,axiom,
    ! [P2: fun_Code_integer_o,Q2: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_ax(P2),Q2)))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P2))
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Q2)) ) ) ).

% finite_Collect_disjI
tff(fact_6060_finite__Collect__conjI,axiom,
    ! [P2: fun_list_nat_o,Q2: fun_list_nat_o] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P2))
        | finite8100373058378681591st_nat(collect_list_nat(Q2)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ay(P2),Q2))) ) ).

% finite_Collect_conjI
tff(fact_6061_finite__Collect__conjI,axiom,
    ! [P2: fun_set_nat_o,Q2: fun_set_nat_o] :
      ( ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P2))
        | aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(Q2)) )
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_az(P2),Q2))) ) ).

% finite_Collect_conjI
tff(fact_6062_finite__Collect__conjI,axiom,
    ! [P2: fun_nat_o,Q2: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(P2))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Q2)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ba(P2),Q2))) ) ).

% finite_Collect_conjI
tff(fact_6063_finite__Collect__conjI,axiom,
    ! [P2: fun_int_o,Q2: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(P2))
        | aa_set_int_o(finite_finite_int,collect_int(Q2)) )
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bb(P2),Q2))) ) ).

% finite_Collect_conjI
tff(fact_6064_finite__Collect__conjI,axiom,
    ! [P2: fun_complex_o,Q2: fun_complex_o] :
      ( ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P2))
        | aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q2)) )
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_bc(P2),Q2))) ) ).

% finite_Collect_conjI
tff(fact_6065_finite__Collect__conjI,axiom,
    ! [P2: fun_Code_integer_o,Q2: fun_Code_integer_o] :
      ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P2))
        | aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Q2)) )
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_bd(P2),Q2))) ) ).

% finite_Collect_conjI
tff(fact_6066_succ__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_succ(T,Xa) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_be(T),Xa)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_6067_pred__empty,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bf(T),Xa)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_6068_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C2: nat,D: nat] :
      ( ( aa_list_nat_nat(size_size_list_nat,Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Ys),I2)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),C2)),aa_list_nat_nat(size_size_list_nat,Ys))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ) ) ) ).

% foldr_mono
tff(fact_6069_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( aa_set_set_nat_o(member_set_nat(I),set_or890127255671739683et_nat(K))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,I),K) ) ).

% lessThan_iff
tff(fact_6070_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( aa_set_rat_o(member_rat(I),set_ord_lessThan_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),K) ) ).

% lessThan_iff
tff(fact_6071_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( aa_set_num_o(member_num(I),set_ord_lessThan_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I),K) ) ).

% lessThan_iff
tff(fact_6072_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% lessThan_iff
tff(fact_6073_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( aa_set_int_o(member_int(I),set_ord_lessThan_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) ) ).

% lessThan_iff
tff(fact_6074_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( aa_set_real_o(member_real(I),set_or5984915006950818249n_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,I),K) ) ).

% lessThan_iff
tff(fact_6075_zip__eq__zip__same__len,axiom,
    ! [A: list_VEBT_VEBT,B: list_VEBT_VEBT,A2: list_VEBT_VEBT,B2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(A) = size_s6755466524823107622T_VEBT(B) )
     => ( ( size_s6755466524823107622T_VEBT(A2) = size_s6755466524823107622T_VEBT(B2) )
       => ( ( zip_VE537291747668921783T_VEBT(A,B) = zip_VE537291747668921783T_VEBT(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6076_zip__eq__zip__same__len,axiom,
    ! [A: list_VEBT_VEBT,B: list_real,A2: list_VEBT_VEBT,B2: list_real] :
      ( ( size_s6755466524823107622T_VEBT(A) = size_size_list_real(B) )
     => ( ( size_s6755466524823107622T_VEBT(A2) = size_size_list_real(B2) )
       => ( ( zip_VEBT_VEBT_real(A,B) = zip_VEBT_VEBT_real(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6077_zip__eq__zip__same__len,axiom,
    ! [A: list_VEBT_VEBT,B: list_o,A2: list_VEBT_VEBT,B2: list_o] :
      ( ( size_s6755466524823107622T_VEBT(A) = size_size_list_o(B) )
     => ( ( size_s6755466524823107622T_VEBT(A2) = size_size_list_o(B2) )
       => ( ( zip_VEBT_VEBT_o(A,B) = zip_VEBT_VEBT_o(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6078_zip__eq__zip__same__len,axiom,
    ! [A: list_VEBT_VEBT,B: list_nat,A2: list_VEBT_VEBT,B2: list_nat] :
      ( ( size_s6755466524823107622T_VEBT(A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( size_s6755466524823107622T_VEBT(A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_VEBT_VEBT_nat(A,B) = zip_VEBT_VEBT_nat(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6079_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_VEBT_VEBT,A2: list_real,B2: list_VEBT_VEBT] :
      ( ( size_size_list_real(A) = size_s6755466524823107622T_VEBT(B) )
     => ( ( size_size_list_real(A2) = size_s6755466524823107622T_VEBT(B2) )
       => ( ( zip_real_VEBT_VEBT(A,B) = zip_real_VEBT_VEBT(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6080_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_real,A2: list_real,B2: list_real] :
      ( ( size_size_list_real(A) = size_size_list_real(B) )
     => ( ( size_size_list_real(A2) = size_size_list_real(B2) )
       => ( ( zip_real_real(A,B) = zip_real_real(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6081_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_o,A2: list_real,B2: list_o] :
      ( ( size_size_list_real(A) = size_size_list_o(B) )
     => ( ( size_size_list_real(A2) = size_size_list_o(B2) )
       => ( ( zip_real_o(A,B) = zip_real_o(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6082_zip__eq__zip__same__len,axiom,
    ! [A: list_real,B: list_nat,A2: list_real,B2: list_nat] :
      ( ( size_size_list_real(A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( size_size_list_real(A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_real_nat(A,B) = zip_real_nat(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6083_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_VEBT_VEBT,A2: list_o,B2: list_VEBT_VEBT] :
      ( ( size_size_list_o(A) = size_s6755466524823107622T_VEBT(B) )
     => ( ( size_size_list_o(A2) = size_s6755466524823107622T_VEBT(B2) )
       => ( ( zip_o_VEBT_VEBT(A,B) = zip_o_VEBT_VEBT(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6084_zip__eq__zip__same__len,axiom,
    ! [A: list_o,B: list_real,A2: list_o,B2: list_real] :
      ( ( size_size_list_o(A) = size_size_list_real(B) )
     => ( ( size_size_list_o(A2) = size_size_list_real(B2) )
       => ( ( zip_o_real(A,B) = zip_o_real(A2,B2) )
        <=> ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_6085_finite__lessThan,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% finite_lessThan
tff(fact_6086_foldr__zero,axiom,
    ! [Xs: list_nat,D: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_list_nat_nat(size_size_list_nat,Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs),I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),D)),D)) ) ).

% foldr_zero
tff(fact_6087_sum_Oneutral__const,axiom,
    ! [A4: set_nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bg),A4) = zero_zero_nat ) ).

% sum.neutral_const
tff(fact_6088_sum_Oneutral__const,axiom,
    ! [A4: set_complex] : ( groups7754918857620584856omplex(aTP_Lamm_bh,A4) = zero_zero_complex ) ).

% sum.neutral_const
tff(fact_6089_sum_Oneutral__const,axiom,
    ! [A4: set_nat] : ( groups6591440286371151544t_real(aTP_Lamm_bi,A4) = zero_zero_real ) ).

% sum.neutral_const
tff(fact_6090_sum_Oneutral__const,axiom,
    ! [A4: set_int] : ( groups4538972089207619220nt_int(aTP_Lamm_bj,A4) = zero_zero_int ) ).

% sum.neutral_const
tff(fact_6091_singleton__conv2,axiom,
    ! [A: vEBT_VEBT] : ( collect_VEBT_VEBT(aa_VEB1521802747257091359VEBT_o(fequal_VEBT_VEBT,A)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ) ).

% singleton_conv2
tff(fact_6092_singleton__conv2,axiom,
    ! [A: complex] : ( collect_complex(aa_com5808801942499186519plex_o(fequal_complex,A)) = insert_complex(A,bot_bot_set_complex) ) ).

% singleton_conv2
tff(fact_6093_singleton__conv2,axiom,
    ! [A: list_nat] : ( collect_list_nat(fequal_list_nat(A)) = insert_list_nat(A,bot_bot_set_list_nat) ) ).

% singleton_conv2
tff(fact_6094_singleton__conv2,axiom,
    ! [A: set_nat] : ( collect_set_nat(aa_set7861427066675100071_nat_o(fequal_set_nat,A)) = aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat) ) ).

% singleton_conv2
tff(fact_6095_singleton__conv2,axiom,
    ! [A: nat] : ( collect_nat(aa_nat_fun_nat_o(fequal_nat,A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ).

% singleton_conv2
tff(fact_6096_singleton__conv2,axiom,
    ! [A: int] : ( collect_int(aa_int_fun_int_o(fequal_int,A)) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) ).

% singleton_conv2
tff(fact_6097_singleton__conv2,axiom,
    ! [A: $o] : ( collect_o(aa_o_fun_o_o(fequal_o,(A))) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) ) ).

% singleton_conv2
tff(fact_6098_singleton__conv,axiom,
    ! [A: vEBT_VEBT] : ( collect_VEBT_VEBT(aa_VEB1521802747257091359VEBT_o(aTP_Lamm_bk,A)) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT) ) ).

% singleton_conv
tff(fact_6099_singleton__conv,axiom,
    ! [A: complex] : ( collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_bl,A)) = insert_complex(A,bot_bot_set_complex) ) ).

% singleton_conv
tff(fact_6100_singleton__conv,axiom,
    ! [A: list_nat] : ( collect_list_nat(aTP_Lamm_bm(A)) = insert_list_nat(A,bot_bot_set_list_nat) ) ).

% singleton_conv
tff(fact_6101_singleton__conv,axiom,
    ! [A: set_nat] : ( collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_bn,A)) = aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat) ) ).

% singleton_conv
tff(fact_6102_singleton__conv,axiom,
    ! [A: nat] : ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bo,A)) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) ) ).

% singleton_conv
tff(fact_6103_singleton__conv,axiom,
    ! [A: int] : ( collect_int(aa_int_fun_int_o(aTP_Lamm_bp,A)) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) ) ).

% singleton_conv
tff(fact_6104_singleton__conv,axiom,
    ! [A: $o] : ( collect_o(aa_o_fun_o_o(aTP_Lamm_bq,(A))) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) ) ).

% singleton_conv
tff(fact_6105_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,K))) ).

% finite_Collect_less_nat
tff(fact_6106_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bs,K))) ).

% finite_Collect_le_nat
tff(fact_6107_card__Collect__less__nat,axiom,
    ! [Na: nat] : ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,Na))) = Na ) ).

% card_Collect_less_nat
tff(fact_6108_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_6109_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_6110_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_6111_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_6112_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = zero_z3403309356797280102nteger )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6113_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_6114_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_6115_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_6116_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_6117_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_6118_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_uint32(semiri2565882477558803405uint32,zero_zero_nat) = zero_zero_uint32 ).

% semiring_1_class.of_nat_0
tff(fact_6119_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% semiring_1_class.of_nat_0
tff(fact_6120_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% semiring_1_class.of_nat_0
tff(fact_6121_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% semiring_1_class.of_nat_0
tff(fact_6122_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% semiring_1_class.of_nat_0
tff(fact_6123_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% semiring_1_class.of_nat_0
tff(fact_6124_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_6125_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_6126_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_6127_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_6128_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_6129_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_6130_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_6131_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_6132_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_6133_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_6134_of__nat__add,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ).

% of_nat_add
tff(fact_6135_of__nat__add,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ).

% of_nat_add
tff(fact_6136_of__nat__add,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ).

% of_nat_add
tff(fact_6137_of__nat__add,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% of_nat_add
tff(fact_6138_of__nat__add,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ) ).

% of_nat_add
tff(fact_6139_of__nat__1,axiom,
    aa_nat_uint32(semiri2565882477558803405uint32,one_one_nat) = one_one_uint32 ).

% of_nat_1
tff(fact_6140_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_6141_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_6142_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_6143_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_6144_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_6145_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6146_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6147_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6148_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6149_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6150_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Na) = one_one_rat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6151_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Na) = one_one_nat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6152_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Na) = one_one_real )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6153_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Na) = one_one_int )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6154_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Na) = one_one_Code_integer )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6155_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_uint32(semiri2565882477558803405uint32,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,M)),aa_nat_uint32(semiri2565882477558803405uint32,Na)) ) ).

% of_nat_mult
tff(fact_6156_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ).

% of_nat_mult
tff(fact_6157_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ).

% of_nat_mult
tff(fact_6158_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% of_nat_mult
tff(fact_6159_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ) ).

% of_nat_mult
tff(fact_6160_lessThan__subset__iff,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_ord_lessThan_rat(Xa)),set_ord_lessThan_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y) ) ).

% lessThan_subset_iff
tff(fact_6161_lessThan__subset__iff,axiom,
    ! [Xa: num,Y: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_ord_lessThan_num(Xa)),set_ord_lessThan_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),Y) ) ).

% lessThan_subset_iff
tff(fact_6162_lessThan__subset__iff,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,Xa)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Y) ) ).

% lessThan_subset_iff
tff(fact_6163_lessThan__subset__iff,axiom,
    ! [Xa: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_lessThan_int(Xa)),set_ord_lessThan_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y) ) ).

% lessThan_subset_iff
tff(fact_6164_lessThan__subset__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or5984915006950818249n_real(Xa)),set_or5984915006950818249n_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% lessThan_subset_iff
tff(fact_6165_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_6166_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups8325533452322294502uint32(aa_fun7720134271958203516uint32(aTP_Lamm_bt(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_6167_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups5944083974425963860uint32(aa_fun39000138016665704uint32(aTP_Lamm_bu(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_6168_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups833757482993574392uint32(aa_fun7322500112117231024uint32(aTP_Lamm_bv(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_6169_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups5712668689793887828uint32(aa_fun7636875859098149992uint32(aTP_Lamm_bw(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_6170_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups8736914816313324502uint32(aa_fun947202747272686700uint32(aTP_Lamm_bx(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_6171_sum_Odelta_H,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups8847630953604152069uint32(aa_fun8111848626357839434uint32(aTP_Lamm_by(A),B),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_Cod8401701657498852871uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta'
tff(fact_6172_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_bz(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6173_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ca(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6174_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_cb(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6175_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_cc(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6176_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups8325533452322294502uint32(aa_fun7720134271958203516uint32(aTP_Lamm_cd(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_6177_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups5944083974425963860uint32(aa_fun39000138016665704uint32(aTP_Lamm_ce(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_6178_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups833757482993574392uint32(aa_fun7322500112117231024uint32(aTP_Lamm_cf(A),B),S2) = $ite(aa_set_nat_o(member_nat(A),S2),aa_nat_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_6179_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups5712668689793887828uint32(aa_fun7636875859098149992uint32(aTP_Lamm_cg(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_6180_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups8736914816313324502uint32(aa_fun947202747272686700uint32(aTP_Lamm_ch(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_6181_sum_Odelta,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups8847630953604152069uint32(aa_fun8111848626357839434uint32(aTP_Lamm_ci(A),B),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_Cod8401701657498852871uint32(B,A),zero_zero_uint32) ) ) ).

% sum.delta
tff(fact_6182_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_cj(A),B),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_VEBT_VEBT_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6183_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ck(A),B),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6184_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_cl(A),B),S2) = $ite(aa_set_int_o(member_int(A),S2),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6185_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_cm(A),B),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6186_foldl__length,axiom,
    ! [L: list_VEBT_VEBT] : ( foldl_nat_VEBT_VEBT(aTP_Lamm_cn,zero_zero_nat,L) = size_s6755466524823107622T_VEBT(L) ) ).

% foldl_length
tff(fact_6187_foldl__length,axiom,
    ! [L: list_real] : ( foldl_nat_real(aTP_Lamm_co,zero_zero_nat,L) = size_size_list_real(L) ) ).

% foldl_length
tff(fact_6188_foldl__length,axiom,
    ! [L: list_o] : ( foldl_nat_o(aTP_Lamm_cp,zero_zero_nat,L) = size_size_list_o(L) ) ).

% foldl_length
tff(fact_6189_foldl__length,axiom,
    ! [L: list_nat] : ( aa_list_nat_nat(foldl_nat_nat(aTP_Lamm_cq,zero_zero_nat),L) = aa_list_nat_nat(size_size_list_nat,L) ) ).

% foldl_length
tff(fact_6190_card__Collect__le__nat,axiom,
    ! [Na: nat] : ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bs,Na))) = aa_nat_nat(suc,Na) ) ).

% card_Collect_le_nat
tff(fact_6191_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_cr,A4))) ) ).

% finite_Collect_subsets
tff(fact_6192_finite__Collect__subsets,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_cs(A4))) ) ).

% finite_Collect_subsets
tff(fact_6193_finite__Collect__subsets,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aTP_Lamm_ct(A4))) ) ).

% finite_Collect_subsets
tff(fact_6194_finite__Collect__subsets,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_cu,A4))) ) ).

% finite_Collect_subsets
tff(fact_6195_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6196_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6197_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6198_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6199_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6200_of__nat__Suc,axiom,
    ! [M: nat] : ( aa_nat_uint32(semiri2565882477558803405uint32,aa_nat_nat(suc,M)) = aa_uint32_uint32(plus_plus_uint32(one_one_uint32),aa_nat_uint32(semiri2565882477558803405uint32,M)) ) ).

% of_nat_Suc
tff(fact_6201_of__nat__Suc,axiom,
    ! [M: nat] : ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ) ).

% of_nat_Suc
tff(fact_6202_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_6203_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_6204_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_6205_of__nat__Suc,axiom,
    ! [M: nat] : ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ) ).

% of_nat_Suc
tff(fact_6206_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_rat(G,Na)) ) ).

% sum.lessThan_Suc
tff(fact_6207_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_int(G,Na)) ) ).

% sum.lessThan_Suc
tff(fact_6208_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_nat(G,Na)) ) ).

% sum.lessThan_Suc
tff(fact_6209_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_real(G,Na)) ) ).

% sum.lessThan_Suc
tff(fact_6210_single__Diff__lessThan,axiom,
    ! [K: $o] : ( minus_minus_set_o(aa_set_o_set_o(insert_o((K)),bot_bot_set_o),set_ord_lessThan_o((K))) = aa_set_o_set_o(insert_o((K)),bot_bot_set_o) ) ).

% single_Diff_lessThan
tff(fact_6211_single__Diff__lessThan,axiom,
    ! [K: nat] : ( minus_minus_set_nat(aa_set_nat_set_nat(insert_nat(K),bot_bot_set_nat),aa_nat_set_nat(set_ord_lessThan_nat,K)) = aa_set_nat_set_nat(insert_nat(K),bot_bot_set_nat) ) ).

% single_Diff_lessThan
tff(fact_6212_single__Diff__lessThan,axiom,
    ! [K: int] : ( minus_minus_set_int(aa_set_int_set_int(insert_int(K),bot_bot_set_int),set_ord_lessThan_int(K)) = aa_set_int_set_int(insert_int(K),bot_bot_set_int) ) ).

% single_Diff_lessThan
tff(fact_6213_single__Diff__lessThan,axiom,
    ! [K: real] : ( minus_minus_set_real(aa_set_real_set_real(insert_real(K),bot_bot_set_real),set_or5984915006950818249n_real(K)) = aa_set_real_set_real(insert_real(K),bot_bot_set_real) ) ).

% single_Diff_lessThan
tff(fact_6214_Cons__listrel1__Cons,axiom,
    ! [Xa: $o,Xs: list_o,Y: $o,Ys: list_o,R: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(cons_o((Xa),Xs),cons_o((Y),Ys)),listrel1_o(R))
    <=> ( ( member7466972457876170832od_o_o(product_Pair_o_o((Xa),(Y)),R)
          & ( Xs = Ys ) )
        | ( ( (Xa)
          <=> (Y) )
          & member4159035015898711888list_o(produc8435520187683070743list_o(Xs,Ys),listrel1_o(R)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_6215_Cons__listrel1__Cons,axiom,
    ! [Xa: uint32,Xs: list_uint32,Y: uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( member2333554998283850498uint32(produc7487160679990061969uint32(cons_uint32(Xa,Xs),cons_uint32(Y,Ys)),listrel1_uint32(R))
    <=> ( ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y),R)
          & ( Xs = Ys ) )
        | ( ( Xa = Y )
          & member2333554998283850498uint32(produc7487160679990061969uint32(Xs,Ys),listrel1_uint32(R)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_6216_Cons__listrel1__Cons,axiom,
    ! [Xa: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(Xa,Xs),cons_nat(Y,Ys)),listrel1_nat(R))
    <=> ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y),R)
          & ( Xs = Ys ) )
        | ( ( Xa = Y )
          & member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_6217_Cons__listrel1__Cons,axiom,
    ! [Xa: int,Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(Xa,Xs),cons_int(Y,Ys)),listrel1_int(R))
    <=> ( ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Y),R)
          & ( Xs = Ys ) )
        | ( ( Xa = Y )
          & member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_6218_zip__Cons__Cons,axiom,
    ! [Xa: int,Xs: list_int,Y: nat,Ys: list_nat] : ( zip_int_nat(cons_int(Xa,Xs),cons_nat(Y,Ys)) = cons_P7512249878480867347nt_nat(product_Pair_int_nat(Xa,Y),zip_int_nat(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6219_zip__Cons__Cons,axiom,
    ! [Xa: int,Xs: list_int,Y: $o,Ys: list_o] : ( zip_int_o(cons_int(Xa,Xs),cons_o((Y),Ys)) = cons_P7321330006258091179_int_o(product_Pair_int_o(Xa,(Y)),zip_int_o(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6220_zip__Cons__Cons,axiom,
    ! [Xa: nat,Xs: list_nat,Y: int,Ys: list_int] : ( zip_nat_int(cons_nat(Xa,Xs),cons_int(Y,Ys)) = cons_P2335045147070616083at_int(product_Pair_nat_int(Xa,Y),zip_nat_int(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6221_zip__Cons__Cons,axiom,
    ! [Xa: nat,Xs: list_nat,Y: $o,Ys: list_o] : ( zip_nat_o(cons_nat(Xa,Xs),cons_o((Y),Ys)) = cons_P9142372351690779143_nat_o(product_Pair_nat_o(Xa,(Y)),zip_nat_o(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6222_zip__Cons__Cons,axiom,
    ! [Xa: $o,Xs: list_o,Y: int,Ys: list_int] : ( zip_o_int(cons_o((Xa),Xs),cons_int(Y,Ys)) = cons_P1455986808126089405_o_int(product_Pair_o_int((Xa),Y),zip_o_int(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6223_zip__Cons__Cons,axiom,
    ! [Xa: $o,Xs: list_o,Y: nat,Ys: list_nat] : ( zip_o_nat(cons_o((Xa),Xs),cons_nat(Y,Ys)) = cons_P5633837827635286113_o_nat(product_Pair_o_nat((Xa),Y),zip_o_nat(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6224_zip__Cons__Cons,axiom,
    ! [Xa: $o,Xs: list_o,Y: $o,Ys: list_o] : ( zip_o_o(cons_o((Xa),Xs),cons_o((Y),Ys)) = cons_P8766293264717362397od_o_o(product_Pair_o_o((Xa),(Y)),zip_o_o(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6225_zip__Cons__Cons,axiom,
    ! [Xa: uint32,Xs: list_uint32,Y: uint32,Ys: list_uint32] : ( zip_uint32_uint32(cons_uint32(Xa,Xs),cons_uint32(Y,Ys)) = cons_P3149448846263281007uint32(produc1400373151660368625uint32(Xa,Y),zip_uint32_uint32(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6226_zip__Cons__Cons,axiom,
    ! [Xa: nat,Xs: list_nat,Y: nat,Ys: list_nat] : ( zip_nat_nat(cons_nat(Xa,Xs),cons_nat(Y,Ys)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y),zip_nat_nat(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6227_zip__Cons__Cons,axiom,
    ! [Xa: int,Xs: list_int,Y: int,Ys: list_int] : ( zip_int_int(cons_int(Xa,Xs),cons_int(Y,Ys)) = cons_P3334398858971670639nt_int(product_Pair_int_int(Xa,Y),zip_int_int(Xs,Ys)) ) ).

% zip_Cons_Cons
tff(fact_6228_Max__const,axiom,
    ! [A4: set_nat,C2: int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( lattic8263393255366662781ax_int(aa_set_nat_set_int(image_nat_int(aTP_Lamm_cv(C2)),A4)) = C2 ) ) ) ).

% Max_const
tff(fact_6229_Max__const,axiom,
    ! [A4: set_int,C2: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( lattic8263393255366662781ax_int(aa_set_int_set_int(image_int_int(aTP_Lamm_cw(C2)),A4)) = C2 ) ) ) ).

% Max_const
tff(fact_6230_Max__const,axiom,
    ! [A4: set_complex,C2: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,image_complex_nat(aTP_Lamm_cx(C2),A4)) = C2 ) ) ) ).

% Max_const
tff(fact_6231_Max__const,axiom,
    ! [A4: set_Code_integer,C2: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,image_951025933927791156er_nat(aTP_Lamm_cy(C2),A4)) = C2 ) ) ) ).

% Max_const
tff(fact_6232_Max__const,axiom,
    ! [A4: set_nat,C2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_cz(C2)),A4)) = C2 ) ) ) ).

% Max_const
tff(fact_6233_Max__const,axiom,
    ! [A4: set_int,C2: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,aa_set_int_set_nat(image_int_nat(aTP_Lamm_da(C2)),A4)) = C2 ) ) ) ).

% Max_const
tff(fact_6234_Max__const,axiom,
    ! [A4: set_o,C2: nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,image_o_nat(aTP_Lamm_db(C2),A4)) = C2 ) ) ) ).

% Max_const
tff(fact_6235_Min__const,axiom,
    ! [A4: set_nat,C2: int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( lattic8718645017227715691in_int(aa_set_nat_set_int(image_nat_int(aTP_Lamm_cv(C2)),A4)) = C2 ) ) ) ).

% Min_const
tff(fact_6236_Min__const,axiom,
    ! [A4: set_int,C2: int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( lattic8718645017227715691in_int(aa_set_int_set_int(image_int_int(aTP_Lamm_cw(C2)),A4)) = C2 ) ) ) ).

% Min_const
tff(fact_6237_Min__const,axiom,
    ! [A4: set_complex,C2: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( ( A4 != bot_bot_set_complex )
       => ( aa_set_nat_nat(lattic8721135487736765967in_nat,image_complex_nat(aTP_Lamm_cx(C2),A4)) = C2 ) ) ) ).

% Min_const
tff(fact_6238_Min__const,axiom,
    ! [A4: set_Code_integer,C2: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( aa_set_nat_nat(lattic8721135487736765967in_nat,image_951025933927791156er_nat(aTP_Lamm_cy(C2),A4)) = C2 ) ) ) ).

% Min_const
tff(fact_6239_Min__const,axiom,
    ! [A4: set_nat,C2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( A4 != bot_bot_set_nat )
       => ( aa_set_nat_nat(lattic8721135487736765967in_nat,aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_cz(C2)),A4)) = C2 ) ) ) ).

% Min_const
tff(fact_6240_Min__const,axiom,
    ! [A4: set_int,C2: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( A4 != bot_bot_set_int )
       => ( aa_set_nat_nat(lattic8721135487736765967in_nat,aa_set_int_set_nat(image_int_nat(aTP_Lamm_da(C2)),A4)) = C2 ) ) ) ).

% Min_const
tff(fact_6241_Min__const,axiom,
    ! [A4: set_o,C2: nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ( A4 != bot_bot_set_o )
       => ( aa_set_nat_nat(lattic8721135487736765967in_nat,image_o_nat(aTP_Lamm_db(C2),A4)) = C2 ) ) ) ).

% Min_const
tff(fact_6242_foldr__length,axiom,
    ! [L: list_VEBT_VEBT] : ( aa_nat_nat(foldr_VEBT_VEBT_nat(aTP_Lamm_dc,L),zero_zero_nat) = size_s6755466524823107622T_VEBT(L) ) ).

% foldr_length
tff(fact_6243_foldr__length,axiom,
    ! [L: list_real] : ( aa_nat_nat(foldr_real_nat(aTP_Lamm_dd,L),zero_zero_nat) = size_size_list_real(L) ) ).

% foldr_length
tff(fact_6244_foldr__length,axiom,
    ! [L: list_o] : ( aa_nat_nat(foldr_o_nat(aTP_Lamm_de,L),zero_zero_nat) = size_size_list_o(L) ) ).

% foldr_length
tff(fact_6245_foldr__length,axiom,
    ! [L: list_nat] : ( aa_nat_nat(foldr_nat_nat(aTP_Lamm_df,L),zero_zero_nat) = aa_list_nat_nat(size_size_list_nat,L) ) ).

% foldr_length
tff(fact_6246_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_6247_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_6248_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_6249_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_6250_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_6251_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6252_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6253_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6254_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6255_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6256_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6257_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6258_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6259_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6260_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6261_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2)),aa_nat_real(semiri5074537144036343181t_real,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6262_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6263_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2)),aa_nat_rat(semiri681578069525770553at_rat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6264_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2)),aa_nat_nat(semiri1316708129612266289at_nat,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6265_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,Xa: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2)),aa_nat_int(semiri1314217659103216013at_int,Xa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)),Xa) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6266_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6267_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6268_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6269_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6270_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xa: nat,B: nat,W2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6271_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6272_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6273_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6274_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6275_of__nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xa)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6276_sum__zero__power,axiom,
    ! [C2: fun_nat_rat,A4: set_nat] :
      ( groups2906978787729119204at_rat(aTP_Lamm_dg(C2),A4) = $ite(
          ( aa_set_nat_o(finite_finite_nat,A4)
          & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
          aa_nat_rat(C2,zero_zero_nat),
          zero_zero_rat ) ) ).

% sum_zero_power
tff(fact_6277_sum__zero__power,axiom,
    ! [C2: fun_nat_complex,A4: set_nat] :
      ( groups2073611262835488442omplex(aTP_Lamm_dh(C2),A4) = $ite(
          ( aa_set_nat_o(finite_finite_nat,A4)
          & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
          aa_nat_complex(C2,zero_zero_nat),
          zero_zero_complex ) ) ).

% sum_zero_power
tff(fact_6278_sum__zero__power,axiom,
    ! [C2: fun_nat_real,A4: set_nat] :
      ( groups6591440286371151544t_real(aTP_Lamm_di(C2),A4) = $ite(
          ( aa_set_nat_o(finite_finite_nat,A4)
          & aa_set_nat_o(member_nat(zero_zero_nat),A4) ),
          aa_nat_real(C2,zero_zero_nat),
          zero_zero_real ) ) ).

% sum_zero_power
tff(fact_6279_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_complex,Na: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_dj(A4),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6280_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_Code_integer,Na: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_dk(A4),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6281_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,Na: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_dl(A4),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6282_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_real,Na: nat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_dm(A4),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6283_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_o,Na: nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_dn(A4),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6284_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_do(A4),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6285_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_int,Na: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_dp(A4),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_6286_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_V854960066525838166emberi(T,Tia,Xa),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_dq(T),Tia),Xa)) ).

% vebt_memberi'_rf_abstr
tff(fact_6287_vebt__succi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_vebt_succi(T,Tia,Xa),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_dr(T),Tia),Xa)) ) ).

% vebt_succi'_rf_abstr
tff(fact_6288_vebt__pred_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_vebt_predi(T,Tia,Xa),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_ds(T),Tia),Xa)) ) ).

% vebt_pred'_rf_abstr
tff(fact_6289_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_V3964819847710782039nserti(T,Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Xa))) ).

% vebt_inserti'_rf_abstr
tff(fact_6290_pred__equals__eq2,axiom,
    ! [R2: set_Pr1773385645901665561uint32,S2: set_Pr1773385645901665561uint32] :
      ( ! [X5: uint32,Xa3: uint32] :
          ( member8027108493173000802uint32(produc1400373151660368625uint32(X5,Xa3),R2)
        <=> member8027108493173000802uint32(produc1400373151660368625uint32(X5,Xa3),S2) )
    <=> ( R2 = S2 ) ) ).

% pred_equals_eq2
tff(fact_6291_pred__equals__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat,Xa3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa3),R2)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa3),S2) )
    <=> ( R2 = S2 ) ) ).

% pred_equals_eq2
tff(fact_6292_pred__equals__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X5: int,Xa3: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(X5,Xa3),R2)
        <=> member5262025264175285858nt_int(product_Pair_int_int(X5,Xa3),S2) )
    <=> ( R2 = S2 ) ) ).

% pred_equals_eq2
tff(fact_6293_pred__equals__eq2,axiom,
    ! [R2: set_Pr1312406793078187350et_nat,S2: set_Pr1312406793078187350et_nat] :
      ( ! [X5: fun_Pr6460944699146269935_nat_o,Xa3: produc3658429121746597890et_nat] :
          ( member4513763875858039863et_nat(produc6017135503889078682et_nat(X5,Xa3),R2)
        <=> member4513763875858039863et_nat(produc6017135503889078682et_nat(X5,Xa3),S2) )
    <=> ( R2 = S2 ) ) ).

% pred_equals_eq2
tff(fact_6294_pred__equals__eq2,axiom,
    ! [R2: set_Pr4040394263813967220et_nat,S2: set_Pr4040394263813967220et_nat] :
      ( ! [X5: fun_Pr6460944699146269935_nat_o,Xa3: produc8799533983703829792et_nat] :
          ( member9174161442226468181et_nat(produc7291855603632349880et_nat(X5,Xa3),R2)
        <=> member9174161442226468181et_nat(produc7291855603632349880et_nat(X5,Xa3),S2) )
    <=> ( R2 = S2 ) ) ).

% pred_equals_eq2
tff(fact_6295_pred__subset__eq,axiom,
    ! [R2: set_nat,S2: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ae(R2),aTP_Lamm_ae(S2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,R2),S2) ) ).

% pred_subset_eq
tff(fact_6296_pred__subset__eq,axiom,
    ! [R2: set_VEBT_VEBT,S2: set_VEBT_VEBT] :
      ( ord_le88303844413636359VEBT_o(aTP_Lamm_a(R2),aTP_Lamm_a(S2))
    <=> aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(R2),S2) ) ).

% pred_subset_eq
tff(fact_6297_pred__subset__eq,axiom,
    ! [R2: set_real,S2: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_aa(R2),aTP_Lamm_aa(S2))
    <=> aa_set_real_o(ord_less_eq_set_real(R2),S2) ) ).

% pred_subset_eq
tff(fact_6298_pred__subset__eq,axiom,
    ! [R2: set_set_nat,S2: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ad(R2),aTP_Lamm_ad(S2))
    <=> aa_set_set_nat_o(ord_le6893508408891458716et_nat(R2),S2) ) ).

% pred_subset_eq
tff(fact_6299_pred__subset__eq,axiom,
    ! [R2: set_int,S2: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_af(R2),aTP_Lamm_af(S2))
    <=> aa_set_int_o(ord_less_eq_set_int(R2),S2) ) ).

% pred_subset_eq
tff(fact_6300_less__eq__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A4),B4)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4)) ) ).

% less_eq_set_def
tff(fact_6301_less__eq__set__def,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A4),B4)
    <=> ord_le88303844413636359VEBT_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4)) ) ).

% less_eq_set_def
tff(fact_6302_less__eq__set__def,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A4),B4)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4)) ) ).

% less_eq_set_def
tff(fact_6303_less__eq__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(A4),B4)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4)) ) ).

% less_eq_set_def
tff(fact_6304_less__eq__set__def,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B4)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_af(A4),aTP_Lamm_af(B4)) ) ).

% less_eq_set_def
tff(fact_6305_Collect__subset,axiom,
    ! [A4: set_VEBT_VEBT,P2: fun_VEBT_VEBT_o] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_dt(A4),P2))),A4) ).

% Collect_subset
tff(fact_6306_Collect__subset,axiom,
    ! [A4: set_real,P2: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_du(A4),P2))),A4) ).

% Collect_subset
tff(fact_6307_Collect__subset,axiom,
    ! [A4: set_complex,P2: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dv(A4),P2))),A4) ).

% Collect_subset
tff(fact_6308_Collect__subset,axiom,
    ! [A4: set_list_nat,P2: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dw(A4),P2)),A4) ).

% Collect_subset
tff(fact_6309_Collect__subset,axiom,
    ! [A4: set_set_nat,P2: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dx(A4),P2))),A4) ).

% Collect_subset
tff(fact_6310_Collect__subset,axiom,
    ! [A4: set_nat,P2: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dy(A4),P2))),A4) ).

% Collect_subset
tff(fact_6311_Collect__subset,axiom,
    ! [A4: set_int,P2: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dz(A4),P2))),A4) ).

% Collect_subset
tff(fact_6312_prop__restrict,axiom,
    ! [Xa: vEBT_VEBT,Z5: set_VEBT_VEBT,X4: set_VEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),Z5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(Z5),collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_dt(X4),P2)))
       => aa_VEBT_VEBT_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_6313_prop__restrict,axiom,
    ! [Xa: real,Z5: set_real,X4: set_real,P2: fun_real_o] :
      ( aa_set_real_o(member_real(Xa),Z5)
     => ( aa_set_real_o(ord_less_eq_set_real(Z5),collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_du(X4),P2)))
       => aa_real_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_6314_prop__restrict,axiom,
    ! [Xa: complex,Z5: set_complex,X4: set_complex,P2: fun_complex_o] :
      ( aa_set_complex_o(member_complex(Xa),Z5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(Z5),collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dv(X4),P2)))
       => aa_complex_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_6315_prop__restrict,axiom,
    ! [Xa: list_nat,Z5: set_list_nat,X4: set_list_nat,P2: fun_list_nat_o] :
      ( member_list_nat(Xa,Z5)
     => ( ord_le6045566169113846134st_nat(Z5,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dw(X4),P2)))
       => aa_list_nat_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_6316_prop__restrict,axiom,
    ! [Xa: set_nat,Z5: set_set_nat,X4: set_set_nat,P2: fun_set_nat_o] :
      ( aa_set_set_nat_o(member_set_nat(Xa),Z5)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(Z5),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dx(X4),P2)))
       => aa_set_nat_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_6317_prop__restrict,axiom,
    ! [Xa: nat,Z5: set_nat,X4: set_nat,P2: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xa),Z5)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z5),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dy(X4),P2)))
       => aa_nat_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_6318_prop__restrict,axiom,
    ! [Xa: int,Z5: set_int,X4: set_int,P2: fun_int_o] :
      ( aa_set_int_o(member_int(Xa),Z5)
     => ( aa_set_int_o(ord_less_eq_set_int(Z5),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dz(X4),P2)))
       => aa_int_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_6319_Collect__restrict,axiom,
    ! [X4: set_VEBT_VEBT,P2: fun_VEBT_VEBT_o] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_dt(X4),P2))),X4) ).

% Collect_restrict
tff(fact_6320_Collect__restrict,axiom,
    ! [X4: set_real,P2: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_du(X4),P2))),X4) ).

% Collect_restrict
tff(fact_6321_Collect__restrict,axiom,
    ! [X4: set_complex,P2: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dv(X4),P2))),X4) ).

% Collect_restrict
tff(fact_6322_Collect__restrict,axiom,
    ! [X4: set_list_nat,P2: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dw(X4),P2)),X4) ).

% Collect_restrict
tff(fact_6323_Collect__restrict,axiom,
    ! [X4: set_set_nat,P2: fun_set_nat_o] : aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dx(X4),P2))),X4) ).

% Collect_restrict
tff(fact_6324_Collect__restrict,axiom,
    ! [X4: set_nat,P2: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dy(X4),P2))),X4) ).

% Collect_restrict
tff(fact_6325_Collect__restrict,axiom,
    ! [X4: set_int,P2: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dz(X4),P2))),X4) ).

% Collect_restrict
tff(fact_6326_subset__CollectI,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,Q2: fun_VEBT_VEBT_o,P2: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),A4)
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),B4)
           => ( aa_VEBT_VEBT_o(Q2,X)
             => aa_VEBT_VEBT_o(P2,X) ) )
       => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_dt(B4),Q2))),collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_dt(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_6327_subset__CollectI,axiom,
    ! [B4: set_real,A4: set_real,Q2: fun_real_o,P2: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(B4),A4)
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),B4)
           => ( aa_real_o(Q2,X)
             => aa_real_o(P2,X) ) )
       => aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_du(B4),Q2))),collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_du(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_6328_subset__CollectI,axiom,
    ! [B4: set_complex,A4: set_complex,Q2: fun_complex_o,P2: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),B4)
           => ( aa_complex_o(Q2,X)
             => aa_complex_o(P2,X) ) )
       => aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dv(B4),Q2))),collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dv(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_6329_subset__CollectI,axiom,
    ! [B4: set_list_nat,A4: set_list_nat,Q2: fun_list_nat_o,P2: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B4,A4)
     => ( ! [X: list_nat] :
            ( member_list_nat(X,B4)
           => ( aa_list_nat_o(Q2,X)
             => aa_list_nat_o(P2,X) ) )
       => ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dw(B4),Q2)),collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dw(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_6330_subset__CollectI,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,Q2: fun_set_nat_o,P2: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B4),A4)
     => ( ! [X: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X),B4)
           => ( aa_set_nat_o(Q2,X)
             => aa_set_nat_o(P2,X) ) )
       => aa_set_set_nat_o(ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dx(B4),Q2))),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dx(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_6331_subset__CollectI,axiom,
    ! [B4: set_nat,A4: set_nat,Q2: fun_nat_o,P2: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B4)
           => ( aa_nat_o(Q2,X)
             => aa_nat_o(P2,X) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dy(B4),Q2))),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dy(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_6332_subset__CollectI,axiom,
    ! [B4: set_int,A4: set_int,Q2: fun_int_o,P2: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B4)
           => ( aa_int_o(Q2,X)
             => aa_int_o(P2,X) ) )
       => aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dz(B4),Q2))),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dz(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_6333_subset__Collect__iff,axiom,
    ! [B4: set_VEBT_VEBT,A4: set_VEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),A4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B4),collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_dt(A4),P2)))
      <=> ! [X5: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),B4)
           => aa_VEBT_VEBT_o(P2,X5) ) ) ) ).

% subset_Collect_iff
tff(fact_6334_subset__Collect__iff,axiom,
    ! [B4: set_real,A4: set_real,P2: fun_real_o] :
      ( aa_set_real_o(ord_less_eq_set_real(B4),A4)
     => ( aa_set_real_o(ord_less_eq_set_real(B4),collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_du(A4),P2)))
      <=> ! [X5: real] :
            ( aa_set_real_o(member_real(X5),B4)
           => aa_real_o(P2,X5) ) ) ) ).

% subset_Collect_iff
tff(fact_6335_subset__Collect__iff,axiom,
    ! [B4: set_complex,A4: set_complex,P2: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dv(A4),P2)))
      <=> ! [X5: complex] :
            ( aa_set_complex_o(member_complex(X5),B4)
           => aa_complex_o(P2,X5) ) ) ) ).

% subset_Collect_iff
tff(fact_6336_subset__Collect__iff,axiom,
    ! [B4: set_list_nat,A4: set_list_nat,P2: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B4,A4)
     => ( ord_le6045566169113846134st_nat(B4,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_dw(A4),P2)))
      <=> ! [X5: list_nat] :
            ( member_list_nat(X5,B4)
           => aa_list_nat_o(P2,X5) ) ) ) ).

% subset_Collect_iff
tff(fact_6337_subset__Collect__iff,axiom,
    ! [B4: set_set_nat,A4: set_set_nat,P2: fun_set_nat_o] :
      ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B4),A4)
     => ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(B4),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_dx(A4),P2)))
      <=> ! [X5: set_nat] :
            ( aa_set_set_nat_o(member_set_nat(X5),B4)
           => aa_set_nat_o(P2,X5) ) ) ) ).

% subset_Collect_iff
tff(fact_6338_subset__Collect__iff,axiom,
    ! [B4: set_nat,A4: set_nat,P2: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dy(A4),P2)))
      <=> ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),B4)
           => aa_nat_o(P2,X5) ) ) ) ).

% subset_Collect_iff
tff(fact_6339_subset__Collect__iff,axiom,
    ! [B4: set_int,A4: set_int,P2: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dz(A4),P2)))
      <=> ! [X5: int] :
            ( aa_set_int_o(member_int(X5),B4)
           => aa_int_o(P2,X5) ) ) ) ).

% subset_Collect_iff
tff(fact_6340_insert__compr,axiom,
    ! [A: $o,B4: set_o] : ( aa_set_o_set_o(insert_o((A)),B4) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_ea((A)),B4)) ) ).

% insert_compr
tff(fact_6341_insert__compr,axiom,
    ! [A: vEBT_VEBT,B4: set_VEBT_VEBT] : ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),B4) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_eb(A),B4)) ) ).

% insert_compr
tff(fact_6342_insert__compr,axiom,
    ! [A: real,B4: set_real] : ( aa_set_real_set_real(insert_real(A),B4) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_ec(A),B4)) ) ).

% insert_compr
tff(fact_6343_insert__compr,axiom,
    ! [A: complex,B4: set_complex] : ( insert_complex(A,B4) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_ed(A),B4)) ) ).

% insert_compr
tff(fact_6344_insert__compr,axiom,
    ! [A: list_nat,B4: set_list_nat] : ( insert_list_nat(A,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_ee(A),B4)) ) ).

% insert_compr
tff(fact_6345_insert__compr,axiom,
    ! [A: set_nat,B4: set_set_nat] : ( aa_set1013257548857649334et_nat(insert_set_nat(A),B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_ef(A),B4)) ) ).

% insert_compr
tff(fact_6346_insert__compr,axiom,
    ! [A: nat,B4: set_nat] : ( aa_set_nat_set_nat(insert_nat(A),B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_eg(A),B4)) ) ).

% insert_compr
tff(fact_6347_insert__compr,axiom,
    ! [A: int,B4: set_int] : ( aa_set_int_set_int(insert_int(A),B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_eh(A),B4)) ) ).

% insert_compr
tff(fact_6348_insert__Collect,axiom,
    ! [A: vEBT_VEBT,P2: fun_VEBT_VEBT_o] : ( aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),collect_VEBT_VEBT(P2)) = collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ei(A),P2)) ) ).

% insert_Collect
tff(fact_6349_insert__Collect,axiom,
    ! [A: $o,P2: fun_o_o] : ( aa_set_o_set_o(insert_o((A)),collect_o(P2)) = collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ej((A)),P2)) ) ).

% insert_Collect
tff(fact_6350_insert__Collect,axiom,
    ! [A: complex,P2: fun_complex_o] : ( insert_complex(A,collect_complex(P2)) = collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ek(A),P2)) ) ).

% insert_Collect
tff(fact_6351_insert__Collect,axiom,
    ! [A: list_nat,P2: fun_list_nat_o] : ( insert_list_nat(A,collect_list_nat(P2)) = collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_el(A),P2)) ) ).

% insert_Collect
tff(fact_6352_insert__Collect,axiom,
    ! [A: set_nat,P2: fun_set_nat_o] : ( aa_set1013257548857649334et_nat(insert_set_nat(A),collect_set_nat(P2)) = collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_em(A),P2)) ) ).

% insert_Collect
tff(fact_6353_insert__Collect,axiom,
    ! [A: nat,P2: fun_nat_o] : ( aa_set_nat_set_nat(insert_nat(A),collect_nat(P2)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_en(A),P2)) ) ).

% insert_Collect
tff(fact_6354_insert__Collect,axiom,
    ! [A: int,P2: fun_int_o] : ( aa_set_int_set_int(insert_int(A),collect_int(P2)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_eo(A),P2)) ) ).

% insert_Collect
tff(fact_6355_Compl__eq,axiom,
    ! [A4: set_VEBT_VEBT] : ( uminus8041839845116263051T_VEBT(A4) = collect_VEBT_VEBT(aTP_Lamm_ep(A4)) ) ).

% Compl_eq
tff(fact_6356_Compl__eq,axiom,
    ! [A4: set_real] : ( uminus612125837232591019t_real(A4) = collect_real(aTP_Lamm_eq(A4)) ) ).

% Compl_eq
tff(fact_6357_Compl__eq,axiom,
    ! [A4: set_complex] : ( uminus8566677241136511917omplex(A4) = collect_complex(aTP_Lamm_er(A4)) ) ).

% Compl_eq
tff(fact_6358_Compl__eq,axiom,
    ! [A4: set_list_nat] : ( uminus3195874150345416415st_nat(A4) = collect_list_nat(aTP_Lamm_es(A4)) ) ).

% Compl_eq
tff(fact_6359_Compl__eq,axiom,
    ! [A4: set_set_nat] : ( uminus613421341184616069et_nat(A4) = collect_set_nat(aTP_Lamm_et(A4)) ) ).

% Compl_eq
tff(fact_6360_Compl__eq,axiom,
    ! [A4: set_nat] : ( uminus5710092332889474511et_nat(A4) = collect_nat(aTP_Lamm_eu(A4)) ) ).

% Compl_eq
tff(fact_6361_Compl__eq,axiom,
    ! [A4: set_int] : ( uminus1532241313380277803et_int(A4) = collect_int(aTP_Lamm_ev(A4)) ) ).

% Compl_eq
tff(fact_6362_Collect__neg__eq,axiom,
    ! [P2: fun_complex_o] : ( collect_complex(aTP_Lamm_ew(P2)) = uminus8566677241136511917omplex(collect_complex(P2)) ) ).

% Collect_neg_eq
tff(fact_6363_Collect__neg__eq,axiom,
    ! [P2: fun_list_nat_o] : ( collect_list_nat(aTP_Lamm_ex(P2)) = uminus3195874150345416415st_nat(collect_list_nat(P2)) ) ).

% Collect_neg_eq
tff(fact_6364_Collect__neg__eq,axiom,
    ! [P2: fun_set_nat_o] : ( collect_set_nat(aTP_Lamm_ey(P2)) = uminus613421341184616069et_nat(collect_set_nat(P2)) ) ).

% Collect_neg_eq
tff(fact_6365_Collect__neg__eq,axiom,
    ! [P2: fun_nat_o] : ( collect_nat(aTP_Lamm_ez(P2)) = uminus5710092332889474511et_nat(collect_nat(P2)) ) ).

% Collect_neg_eq
tff(fact_6366_Collect__neg__eq,axiom,
    ! [P2: fun_int_o] : ( collect_int(aTP_Lamm_fa(P2)) = uminus1532241313380277803et_int(collect_int(P2)) ) ).

% Collect_neg_eq
tff(fact_6367_uminus__set__def,axiom,
    ! [A4: set_VEBT_VEBT] : ( uminus8041839845116263051T_VEBT(A4) = collect_VEBT_VEBT(uminus3031588813561569520VEBT_o(aTP_Lamm_a(A4))) ) ).

% uminus_set_def
tff(fact_6368_uminus__set__def,axiom,
    ! [A4: set_real] : ( uminus612125837232591019t_real(A4) = collect_real(uminus2572480474538279996real_o(aTP_Lamm_aa(A4))) ) ).

% uminus_set_def
tff(fact_6369_uminus__set__def,axiom,
    ! [A4: set_complex] : ( uminus8566677241136511917omplex(A4) = collect_complex(uminus4035047602201070266plex_o(aTP_Lamm_ab(A4))) ) ).

% uminus_set_def
tff(fact_6370_uminus__set__def,axiom,
    ! [A4: set_list_nat] : ( uminus3195874150345416415st_nat(A4) = collect_list_nat(uminus8676031017987022600_nat_o(aTP_Lamm_ac(A4))) ) ).

% uminus_set_def
tff(fact_6371_uminus__set__def,axiom,
    ! [A4: set_set_nat] : ( uminus613421341184616069et_nat(A4) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_ad(A4))) ) ).

% uminus_set_def
tff(fact_6372_uminus__set__def,axiom,
    ! [A4: set_nat] : ( uminus5710092332889474511et_nat(A4) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ae(A4))) ) ).

% uminus_set_def
tff(fact_6373_uminus__set__def,axiom,
    ! [A4: set_int] : ( uminus1532241313380277803et_int(A4) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_af(A4))) ) ).

% uminus_set_def
tff(fact_6374_lessThan__def,axiom,
    ! [U: set_nat] : ( set_or890127255671739683et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_fb,U)) ) ).

% lessThan_def
tff(fact_6375_lessThan__def,axiom,
    ! [U: rat] : ( set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_fc,U)) ) ).

% lessThan_def
tff(fact_6376_lessThan__def,axiom,
    ! [U: num] : ( set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_fd,U)) ) ).

% lessThan_def
tff(fact_6377_lessThan__def,axiom,
    ! [U: nat] : ( aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,U)) ) ).

% lessThan_def
tff(fact_6378_lessThan__def,axiom,
    ! [U: int] : ( set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_fe,U)) ) ).

% lessThan_def
tff(fact_6379_lessThan__def,axiom,
    ! [U: real] : ( set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_ff,U)) ) ).

% lessThan_def
tff(fact_6380_less__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A4),B4)
    <=> ord_less_fun_nat_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4)) ) ).

% less_set_def
tff(fact_6381_less__set__def,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A4),B4)
    <=> ord_le3534172602983796987VEBT_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4)) ) ).

% less_set_def
tff(fact_6382_less__set__def,axiom,
    ! [A4: set_real,B4: set_real] :
      ( aa_set_real_o(ord_less_set_real(A4),B4)
    <=> ord_less_fun_real_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4)) ) ).

% less_set_def
tff(fact_6383_less__set__def,axiom,
    ! [A4: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A4),B4)
    <=> ord_less_fun_int_o(aTP_Lamm_af(A4),aTP_Lamm_af(B4)) ) ).

% less_set_def
tff(fact_6384_less__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( aa_set_set_nat_o(ord_less_set_set_nat(A4),B4)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4)) ) ).

% less_set_def
tff(fact_6385_Set_Oempty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_fg) ).

% Set.empty_def
tff(fact_6386_Set_Oempty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_fh) ).

% Set.empty_def
tff(fact_6387_Set_Oempty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_fi) ).

% Set.empty_def
tff(fact_6388_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_fj) ).

% Set.empty_def
tff(fact_6389_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_fk) ).

% Set.empty_def
tff(fact_6390_Set_Oempty__def,axiom,
    bot_bot_set_o = collect_o(aTP_Lamm_fl) ).

% Set.empty_def
tff(fact_6391_minus__set__def,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] : ( minus_5127226145743854075T_VEBT(A4,B4) = collect_VEBT_VEBT(minus_4582106842932979072VEBT_o(aTP_Lamm_a(A4),aTP_Lamm_a(B4))) ) ).

% minus_set_def
tff(fact_6392_minus__set__def,axiom,
    ! [A4: set_real,B4: set_real] : ( minus_minus_set_real(A4,B4) = collect_real(minus_404528632214381484real_o(aTP_Lamm_aa(A4),aTP_Lamm_aa(B4))) ) ).

% minus_set_def
tff(fact_6393_minus__set__def,axiom,
    ! [A4: set_complex,B4: set_complex] : ( minus_811609699411566653omplex(A4,B4) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_ab(A4),aTP_Lamm_ab(B4))) ) ).

% minus_set_def
tff(fact_6394_minus__set__def,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : ( minus_7954133019191499631st_nat(A4,B4) = collect_list_nat(minus_5761417318614613624_nat_o(aTP_Lamm_ac(A4),aTP_Lamm_ac(B4))) ) ).

% minus_set_def
tff(fact_6395_minus__set__def,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : ( minus_2163939370556025621et_nat(A4,B4) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ad(A4),aTP_Lamm_ad(B4))) ) ).

% minus_set_def
tff(fact_6396_minus__set__def,axiom,
    ! [A4: set_int,B4: set_int] : ( minus_minus_set_int(A4,B4) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_af(A4),aTP_Lamm_af(B4))) ) ).

% minus_set_def
tff(fact_6397_minus__set__def,axiom,
    ! [A4: set_nat,B4: set_nat] : ( minus_minus_set_nat(A4,B4) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ae(A4),aTP_Lamm_ae(B4))) ) ).

% minus_set_def
tff(fact_6398_set__diff__eq,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_VEBT_VEBT] : ( minus_5127226145743854075T_VEBT(A4,B4) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_fm(A4),B4)) ) ).

% set_diff_eq
tff(fact_6399_set__diff__eq,axiom,
    ! [A4: set_real,B4: set_real] : ( minus_minus_set_real(A4,B4) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_fn(A4),B4)) ) ).

% set_diff_eq
tff(fact_6400_set__diff__eq,axiom,
    ! [A4: set_complex,B4: set_complex] : ( minus_811609699411566653omplex(A4,B4) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_fo(A4),B4)) ) ).

% set_diff_eq
tff(fact_6401_set__diff__eq,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] : ( minus_7954133019191499631st_nat(A4,B4) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_fp(A4),B4)) ) ).

% set_diff_eq
tff(fact_6402_set__diff__eq,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] : ( minus_2163939370556025621et_nat(A4,B4) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_fq(A4),B4)) ) ).

% set_diff_eq
tff(fact_6403_set__diff__eq,axiom,
    ! [A4: set_int,B4: set_int] : ( minus_minus_set_int(A4,B4) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_fr(A4),B4)) ) ).

% set_diff_eq
tff(fact_6404_set__diff__eq,axiom,
    ! [A4: set_nat,B4: set_nat] : ( minus_minus_set_nat(A4,B4) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_fs(A4),B4)) ) ).

% set_diff_eq
tff(fact_6405_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_nat,R2: fun_VE6480751511521678896_nat_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
             => ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),B4)
                  & aa_nat_o(aa_VEB2314620471664397097_nat_o(R2,X),Xa2) ) )
         => ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),B4)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_ft(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6406_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_nat,R2: fun_real_fun_nat_o] :
      ( ~ aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),A4)
             => ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),B4)
                  & aa_nat_o(aa_real_fun_nat_o(R2,X),Xa2) ) )
         => ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),B4)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_fu(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6407_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_int,R2: fun_VE4235606543851232980_int_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
             => ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),B4)
                  & aa_int_o(aa_VEB493578126231709133_int_o(R2,X),Xa2) ) )
         => ? [X: int] :
              ( aa_set_int_o(member_int(X),B4)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_int8839263053405020433VEBT_o(aa_fun8807180201667926370VEBT_o(aTP_Lamm_fv(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6408_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_int,R2: fun_real_fun_int_o] :
      ( ~ aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),A4)
             => ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),B4)
                  & aa_int_o(aa_real_fun_int_o(R2,X),Xa2) ) )
         => ? [X: int] :
              ( aa_set_int_o(member_int(X),B4)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_fw(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6409_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_complex,R2: fun_VE7631951351884037842plex_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
             => ? [Xa2: complex] :
                  ( aa_set_complex_o(member_complex(Xa2),B4)
                  & aa_complex_o(aa_VEB8036846806599048395plex_o(R2,X),Xa2) ) )
         => ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),B4)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_com3651963455866140051VEBT_o(aa_fun7268111043052431330VEBT_o(aTP_Lamm_fx(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6410_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_complex,R2: fun_re7319081385389557872plex_o] :
      ( ~ aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),A4)
             => ? [Xa2: complex] :
                  ( aa_set_complex_o(member_complex(Xa2),B4)
                  & aa_complex_o(aa_rea2382570592068293209plex_o(R2,X),Xa2) ) )
         => ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),B4)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_fy(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6411_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_Code_integer,R2: fun_VE1909044406494083811eger_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
             => ? [Xa2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),B4)
                  & aa_Code_integer_o(aa_VEB1409644639000594780eger_o(R2,X),Xa2) ) )
         => ? [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),B4)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_Cod5040069908416755522VEBT_o(aa_fun1637734041313256866VEBT_o(aTP_Lamm_fz(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6412_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B4: set_Code_integer,R2: fun_re3424108668441675905eger_o] :
      ( ~ aa_set_real_o(finite_finite_real,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B4)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),A4)
             => ? [Xa2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa2),B4)
                  & aa_Code_integer_o(aa_rea4090721512283939562eger_o(R2,X),Xa2) ) )
         => ? [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),B4)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_Cod1315720040876898922real_o(aa_fun4247632250071955964real_o(aTP_Lamm_ga(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6413_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_nat,R2: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
             => ? [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(R2,X),Xa2) ) )
         => ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_gb(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6414_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_int,R2: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A4)
             => ? [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),B4)
                  & aa_int_o(aa_nat_fun_int_o(R2,X),Xa2) ) )
         => ? [X: int] :
              ( aa_set_int_o(member_int(X),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_gc(A4),R2),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_6415_not__finite__existsD,axiom,
    ! [P2: fun_list_nat_o] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P2))
     => ? [X_1: list_nat] : aa_list_nat_o(P2,X_1) ) ).

% not_finite_existsD
tff(fact_6416_not__finite__existsD,axiom,
    ! [P2: fun_set_nat_o] :
      ( ~ aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(P2))
     => ? [X_1: set_nat] : aa_set_nat_o(P2,X_1) ) ).

% not_finite_existsD
tff(fact_6417_not__finite__existsD,axiom,
    ! [P2: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(P2))
     => ? [X_1: nat] : aa_nat_o(P2,X_1) ) ).

% not_finite_existsD
tff(fact_6418_not__finite__existsD,axiom,
    ! [P2: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(P2))
     => ? [X_1: int] : aa_int_o(P2,X_1) ) ).

% not_finite_existsD
tff(fact_6419_not__finite__existsD,axiom,
    ! [P2: fun_complex_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P2))
     => ? [X_1: complex] : aa_complex_o(P2,X_1) ) ).

% not_finite_existsD
tff(fact_6420_not__finite__existsD,axiom,
    ! [P2: fun_Code_integer_o] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P2))
     => ? [X_1: code_integer] : aa_Code_integer_o(P2,X_1) ) ).

% not_finite_existsD
tff(fact_6421_imageE,axiom,
    ! [B: nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ~ ! [X: nat] :
            ( ( B = aa_nat_nat(F,X) )
           => ~ aa_set_nat_o(member_nat(X),A4) ) ) ).

% imageE
tff(fact_6422_imageE,axiom,
    ! [B: nat,F: fun_VEBT_VEBT_nat,A4: set_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(B),image_VEBT_VEBT_nat(F,A4))
     => ~ ! [X: vEBT_VEBT] :
            ( ( B = aa_VEBT_VEBT_nat(F,X) )
           => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4) ) ) ).

% imageE
tff(fact_6423_imageE,axiom,
    ! [B: nat,F: fun_real_nat,A4: set_real] :
      ( aa_set_nat_o(member_nat(B),image_real_nat(F,A4))
     => ~ ! [X: real] :
            ( ( B = aa_real_nat(F,X) )
           => ~ aa_set_real_o(member_real(X),A4) ) ) ).

% imageE
tff(fact_6424_imageE,axiom,
    ! [B: nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(member_nat(B),aa_set_int_set_nat(image_int_nat(F),A4))
     => ~ ! [X: int] :
            ( ( B = aa_int_nat(F,X) )
           => ~ aa_set_int_o(member_int(X),A4) ) ) ).

% imageE
tff(fact_6425_imageE,axiom,
    ! [B: vEBT_VEBT,F: fun_nat_VEBT_VEBT,A4: set_nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_nat_VEBT_VEBT(F,A4))
     => ~ ! [X: nat] :
            ( ( B = aa_nat_VEBT_VEBT(F,X) )
           => ~ aa_set_nat_o(member_nat(X),A4) ) ) ).

% imageE
tff(fact_6426_imageE,axiom,
    ! [B: vEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,A4: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_3375948659692109573T_VEBT(F,A4))
     => ~ ! [X: vEBT_VEBT] :
            ( ( B = aa_VEB5832953974749077948T_VEBT(F,X) )
           => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4) ) ) ).

% imageE
tff(fact_6427_imageE,axiom,
    ! [B: vEBT_VEBT,F: fun_real_VEBT_VEBT,A4: set_real] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_real_VEBT_VEBT(F,A4))
     => ~ ! [X: real] :
            ( ( B = aa_real_VEBT_VEBT(F,X) )
           => ~ aa_set_real_o(member_real(X),A4) ) ) ).

% imageE
tff(fact_6428_imageE,axiom,
    ! [B: vEBT_VEBT,F: fun_int_VEBT_VEBT,A4: set_int] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),image_int_VEBT_VEBT(F,A4))
     => ~ ! [X: int] :
            ( ( B = aa_int_VEBT_VEBT(F,X) )
           => ~ aa_set_int_o(member_int(X),A4) ) ) ).

% imageE
tff(fact_6429_imageE,axiom,
    ! [B: real,F: fun_nat_real,A4: set_nat] :
      ( aa_set_real_o(member_real(B),image_nat_real(F,A4))
     => ~ ! [X: nat] :
            ( ( B = aa_nat_real(F,X) )
           => ~ aa_set_nat_o(member_nat(X),A4) ) ) ).

% imageE
tff(fact_6430_imageE,axiom,
    ! [B: real,F: fun_VEBT_VEBT_real,A4: set_VEBT_VEBT] :
      ( aa_set_real_o(member_real(B),image_VEBT_VEBT_real(F,A4))
     => ~ ! [X: vEBT_VEBT] :
            ( ( B = aa_VEBT_VEBT_real(F,X) )
           => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4) ) ) ).

% imageE
tff(fact_6431_image__image,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A4: set_nat] : ( aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(image_nat_nat(G),A4)) = aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gd(F),G)),A4) ) ).

% image_image
tff(fact_6432_image__image,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,A4: set_int] : ( aa_set_nat_set_nat(image_nat_nat(F),aa_set_int_set_nat(image_int_nat(G),A4)) = aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ge(F),G)),A4) ) ).

% image_image
tff(fact_6433_image__image,axiom,
    ! [F: fun_nat_int,G: fun_nat_nat,A4: set_nat] : ( aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(image_nat_nat(G),A4)) = aa_set_nat_set_int(image_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_gf(F),G)),A4) ) ).

% image_image
tff(fact_6434_image__image,axiom,
    ! [F: fun_nat_int,G: fun_int_nat,A4: set_int] : ( aa_set_nat_set_int(image_nat_int(F),aa_set_int_set_nat(image_int_nat(G),A4)) = aa_set_int_set_int(image_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_gg(F),G)),A4) ) ).

% image_image
tff(fact_6435_image__image,axiom,
    ! [F: fun_int_nat,G: fun_nat_int,A4: set_nat] : ( aa_set_int_set_nat(image_int_nat(F),aa_set_nat_set_int(image_nat_int(G),A4)) = aa_set_nat_set_nat(image_nat_nat(aa_fun519522104774525396at_nat(aTP_Lamm_gh(F),G)),A4) ) ).

% image_image
tff(fact_6436_image__image,axiom,
    ! [F: fun_int_nat,G: fun_int_int,A4: set_int] : ( aa_set_int_set_nat(image_int_nat(F),aa_set_int_set_int(image_int_int(G),A4)) = aa_set_int_set_nat(image_int_nat(aa_fun1234824742305470092nt_nat(aTP_Lamm_gi(F),G)),A4) ) ).

% image_image
tff(fact_6437_image__image,axiom,
    ! [F: fun_int_int,G: fun_nat_int,A4: set_nat] : ( aa_set_int_set_int(image_int_int(F),aa_set_nat_set_int(image_nat_int(G),A4)) = aa_set_nat_set_int(image_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_gj(F),G)),A4) ) ).

% image_image
tff(fact_6438_image__image,axiom,
    ! [F: fun_int_int,G: fun_int_int,A4: set_int] : ( aa_set_int_set_int(image_int_int(F),aa_set_int_set_int(image_int_int(G),A4)) = aa_set_int_set_int(image_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gk(F),G)),A4) ) ).

% image_image
tff(fact_6439_image__image,axiom,
    ! [F: fun_set_nat_nat,G: fun_nat_set_nat,A4: set_nat] : ( image_set_nat_nat(F,aa_set7246812951530217472et_nat(image_nat_set_nat(G),A4)) = aa_set_nat_set_nat(image_nat_nat(aa_fun4820382421437094446at_nat(aTP_Lamm_gl(F),G)),A4) ) ).

% image_image
tff(fact_6440_image__image,axiom,
    ! [F: fun_set_nat_int,G: fun_nat_set_nat,A4: set_nat] : ( image_set_nat_int(F,aa_set7246812951530217472et_nat(image_nat_set_nat(G),A4)) = aa_set_nat_set_int(image_nat_int(aa_fun642531401927897738at_int(aTP_Lamm_gm(F),G)),A4) ) ).

% image_image
tff(fact_6441_Compr__image__eq,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,A4: set_VEBT_VEBT,P2: fun_VEBT_VEBT_o] : ( collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aa_set7749190676507875235VEBT_o(aTP_Lamm_gn(F),A4),P2)) = image_3375948659692109573T_VEBT(F,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aa_set7749190676507875235VEBT_o(aTP_Lamm_go(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6442_Compr__image__eq,axiom,
    ! [F: fun_real_VEBT_VEBT,A4: set_real,P2: fun_VEBT_VEBT_o] : ( collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aa_set6460758714441890155VEBT_o(aTP_Lamm_gp(F),A4),P2)) = image_real_VEBT_VEBT(F,collect_real(aa_fun2752195911241646752real_o(aa_set5716891373128775959real_o(aTP_Lamm_gq(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6443_Compr__image__eq,axiom,
    ! [F: fun_VEBT_VEBT_real,A4: set_VEBT_VEBT,P2: fun_real_o] : ( collect_real(aa_fun5868358341814070780real_o(aa_set5216747956267341787real_o(aTP_Lamm_gr(F),A4),P2)) = image_VEBT_VEBT_real(F,collect_VEBT_VEBT(aa_fun7216158613899282224VEBT_o(aa_set4022976891253242279VEBT_o(aTP_Lamm_gs(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6444_Compr__image__eq,axiom,
    ! [F: fun_real_real,A4: set_real,P2: fun_real_o] : ( collect_real(aa_fun5868358341814070780real_o(aa_set1691088162573342115real_o(aTP_Lamm_gt(F),A4),P2)) = image_real_real(F,collect_real(aa_fun5868358341814070780real_o(aa_set1691088162573342115real_o(aTP_Lamm_gu(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6445_Compr__image__eq,axiom,
    ! [F: fun_co4340422964966740447T_VEBT,A4: set_complex,P2: fun_VEBT_VEBT_o] : ( collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aa_set5809115984592220781VEBT_o(aTP_Lamm_gv(F),A4),P2)) = image_932796090930683071T_VEBT(F,collect_complex(aa_fun7745313613784576798plex_o(aa_set1943973126016318227plex_o(aTP_Lamm_gw(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6446_Compr__image__eq,axiom,
    ! [F: fun_complex_real,A4: set_complex,P2: fun_real_o] : ( collect_real(aa_fun5868358341814070780real_o(aa_set8509023117188218277real_o(aTP_Lamm_gx(F),A4),P2)) = image_complex_real(F,collect_complex(aa_fun3391577472191272058plex_o(aa_set4195536971368458275plex_o(aTP_Lamm_gy(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6447_Compr__image__eq,axiom,
    ! [F: fun_nat_VEBT_VEBT,A4: set_nat,P2: fun_VEBT_VEBT_o] : ( collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aa_set2181257101989441679VEBT_o(aTP_Lamm_gz(F),A4),P2)) = image_nat_VEBT_VEBT(F,collect_nat(aa_fun370934926479296380_nat_o(aa_set5037496533744297423_nat_o(aTP_Lamm_ha(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6448_Compr__image__eq,axiom,
    ! [F: fun_nat_real,A4: set_nat,P2: fun_real_o] : ( collect_real(aa_fun5868358341814070780real_o(aa_set8653532325685533383real_o(aTP_Lamm_hb(F),A4),P2)) = image_nat_real(F,collect_nat(aa_fun966921962273399000_nat_o(aa_set47184369695174563_nat_o(aTP_Lamm_hc(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6449_Compr__image__eq,axiom,
    ! [F: fun_int_VEBT_VEBT,A4: set_int,P2: fun_VEBT_VEBT_o] : ( collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aa_set2625704858979348715VEBT_o(aTP_Lamm_hd(F),A4),P2)) = image_int_VEBT_VEBT(F,collect_int(aa_fun7773264617901384224_int_o(aa_set2765448116205184791_int_o(aTP_Lamm_he(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6450_Compr__image__eq,axiom,
    ! [F: fun_int_real,A4: set_int,P2: fun_real_o] : ( collect_real(aa_fun5868358341814070780real_o(aa_set7127480517943840035real_o(aTP_Lamm_hf(F),A4),P2)) = image_int_real(F,collect_int(aa_fun8369251653695486844_int_o(aa_set5431007883013265571_int_o(aTP_Lamm_hg(F),A4),P2))) ) ).

% Compr_image_eq
tff(fact_6451_lambda__zero,axiom,
    aTP_Lamm_hh = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_6452_lambda__zero,axiom,
    aTP_Lamm_hi = aa_uin3778660748068847871uint32(times_times_uint32,zero_zero_uint32) ).

% lambda_zero
tff(fact_6453_lambda__zero,axiom,
    aTP_Lamm_hj = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_6454_lambda__zero,axiom,
    aTP_Lamm_bg = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_6455_lambda__zero,axiom,
    aTP_Lamm_bj = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_6456_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Na: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hk(G),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ) ).

% sum.nat_diff_reindex
tff(fact_6457_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Na: nat] : ( groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_hl(G),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,Na)) ) ).

% sum.nat_diff_reindex
tff(fact_6458_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,K4)),groups2906978787729119204at_rat(G,K4)) ) ).

% sum_mono
tff(fact_6459_sum__mono,axiom,
    ! [K4: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),K4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I2)),aa_VEBT_VEBT_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups136491112297645522BT_rat(F,K4)),groups136491112297645522BT_rat(G,K4)) ) ).

% sum_mono
tff(fact_6460_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),K4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,K4)),groups1300246762558778688al_rat(G,K4)) ) ).

% sum_mono
tff(fact_6461_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K4)),groups3906332499630173760nt_rat(G,K4)) ) ).

% sum_mono
tff(fact_6462_sum__mono,axiom,
    ! [K4: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),K4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I2)),aa_VEBT_VEBT_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups771621172384141258BT_nat(F,K4)),groups771621172384141258BT_nat(G,K4)) ) ).

% sum_mono
tff(fact_6463_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),K4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K4)),groups1935376822645274424al_nat(G,K4)) ) ).

% sum_mono
tff(fact_6464_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K4)),groups4541462559716669496nt_nat(G,K4)) ) ).

% sum_mono
tff(fact_6465_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_nat_int(groups3539618377306564664at_int(F),K4)),aa_set_nat_int(groups3539618377306564664at_int(G),K4)) ) ).

% sum_mono
tff(fact_6466_sum__mono,axiom,
    ! [K4: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),K4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_VEBT_VEBT_int(F,I2)),aa_VEBT_VEBT_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups769130701875090982BT_int(F,K4)),groups769130701875090982BT_int(G,K4)) ) ).

% sum_mono
tff(fact_6467_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),K4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K4)),groups1932886352136224148al_int(G,K4)) ) ).

% sum_mono
tff(fact_6468_pigeonhole__infinite,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_nat_o(finite_finite_nat,image_VEBT_VEBT_nat(F,A4))
       => ? [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
            & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_VEB1521802747257091359VEBT_o(aa_fun8512903131112361427VEBT_o(aTP_Lamm_hm(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6469_pigeonhole__infinite,axiom,
    ! [A4: set_real,F: fun_real_nat] :
      ( ~ aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_nat_o(finite_finite_nat,image_real_nat(F,A4))
       => ? [X: real] :
            ( aa_set_real_o(member_real(X),A4)
            & ~ aa_set_real_o(finite_finite_real,collect_real(aa_real_fun_real_o(aa_fun8229752488527876603real_o(aTP_Lamm_hn(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6470_pigeonhole__infinite,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_int_o(finite_finite_int,image_VEBT_VEBT_int(F,A4))
       => ? [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
            & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_VEB1521802747257091359VEBT_o(aa_fun211421702454883447VEBT_o(aTP_Lamm_ho(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6471_pigeonhole__infinite,axiom,
    ! [A4: set_real,F: fun_real_int] :
      ( ~ aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_int_o(finite_finite_int,image_real_int(F,A4))
       => ? [X: real] :
            ( aa_set_real_o(member_real(X),A4)
            & ~ aa_set_real_o(finite_finite_real,collect_real(aa_real_fun_real_o(aa_fun7132669507883878047real_o(aTP_Lamm_hp(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6472_pigeonhole__infinite,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VE5569070535936029429omplex] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,image_3793382806556112285omplex(F,A4))
       => ? [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
            & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_VEB1521802747257091359VEBT_o(aa_fun7842805400471075189VEBT_o(aTP_Lamm_hq(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6473_pigeonhole__infinite,axiom,
    ! [A4: set_real,F: fun_real_complex] :
      ( ~ aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,image_real_complex(F,A4))
       => ? [X: real] :
            ( aa_set_real_o(member_real(X),A4)
            & ~ aa_set_real_o(finite_finite_real,collect_real(aa_real_fun_real_o(aa_fun3021595049966113949real_o(aTP_Lamm_hr(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6474_pigeonhole__infinite,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_2092689629700589388nteger(F,A4))
       => ? [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
            & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_VEB1521802747257091359VEBT_o(aa_fun8260802627938895110VEBT_o(aTP_Lamm_hs(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6475_pigeonhole__infinite,axiom,
    ! [A4: set_real,F: fun_re6978948838062957830nteger] :
      ( ~ aa_set_real_o(finite_finite_real,A4)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,image_4958697645175560720nteger(F,A4))
       => ? [X: real] :
            ( aa_set_real_o(member_real(X),A4)
            & ~ aa_set_real_o(finite_finite_real,collect_real(aa_real_fun_real_o(aa_fun7085772160498067374real_o(aTP_Lamm_ht(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6476_pigeonhole__infinite,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_hu(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6477_pigeonhole__infinite,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun4569312790843823667_nat_o(aTP_Lamm_hv(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6478_image__Collect__subsetI,axiom,
    ! [P2: fun_complex_o,F: fun_complex_nat,B4: set_nat] :
      ( ! [X: complex] :
          ( aa_complex_o(P2,X)
         => aa_set_nat_o(member_nat(aa_complex_nat(F,X)),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_complex_nat(F,collect_complex(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6479_image__Collect__subsetI,axiom,
    ! [P2: fun_complex_o,F: fun_co4340422964966740447T_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X: complex] :
          ( aa_complex_o(P2,X)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_complex_VEBT_VEBT(F,X)),B4) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(image_932796090930683071T_VEBT(F,collect_complex(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6480_image__Collect__subsetI,axiom,
    ! [P2: fun_complex_o,F: fun_complex_real,B4: set_real] :
      ( ! [X: complex] :
          ( aa_complex_o(P2,X)
         => aa_set_real_o(member_real(aa_complex_real(F,X)),B4) )
     => aa_set_real_o(ord_less_eq_set_real(image_complex_real(F,collect_complex(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6481_image__Collect__subsetI,axiom,
    ! [P2: fun_nat_o,F: fun_nat_nat,B4: set_nat] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X)),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(F),collect_nat(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6482_image__Collect__subsetI,axiom,
    ! [P2: fun_nat_o,F: fun_nat_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_nat_VEBT_VEBT(F,X)),B4) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(image_nat_VEBT_VEBT(F,collect_nat(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6483_image__Collect__subsetI,axiom,
    ! [P2: fun_nat_o,F: fun_nat_real,B4: set_real] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
         => aa_set_real_o(member_real(aa_nat_real(F,X)),B4) )
     => aa_set_real_o(ord_less_eq_set_real(image_nat_real(F,collect_nat(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6484_image__Collect__subsetI,axiom,
    ! [P2: fun_int_o,F: fun_int_nat,B4: set_nat] :
      ( ! [X: int] :
          ( aa_int_o(P2,X)
         => aa_set_nat_o(member_nat(aa_int_nat(F,X)),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_int_set_nat(image_int_nat(F),collect_int(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6485_image__Collect__subsetI,axiom,
    ! [P2: fun_int_o,F: fun_int_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ! [X: int] :
          ( aa_int_o(P2,X)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_int_VEBT_VEBT(F,X)),B4) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(image_int_VEBT_VEBT(F,collect_int(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6486_image__Collect__subsetI,axiom,
    ! [P2: fun_int_o,F: fun_int_real,B4: set_real] :
      ( ! [X: int] :
          ( aa_int_o(P2,X)
         => aa_set_real_o(member_real(aa_int_real(F,X)),B4) )
     => aa_set_real_o(ord_less_eq_set_real(image_int_real(F,collect_int(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6487_image__Collect__subsetI,axiom,
    ! [P2: fun_complex_o,F: fun_complex_int,B4: set_int] :
      ( ! [X: complex] :
          ( aa_complex_o(P2,X)
         => aa_set_int_o(member_int(aa_complex_int(F,X)),B4) )
     => aa_set_int_o(ord_less_eq_set_int(image_complex_int(F,collect_complex(P2))),B4) ) ).

% image_Collect_subsetI
tff(fact_6488_Collect__conv__if2,axiom,
    ! [A: vEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_hw(A),P2)) = $ite(aa_VEBT_VEBT_o(P2,A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ) ).

% Collect_conv_if2
tff(fact_6489_Collect__conv__if2,axiom,
    ! [A: complex,P2: fun_complex_o] :
      ( collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_hx(A),P2)) = $ite(aa_complex_o(P2,A),insert_complex(A,bot_bot_set_complex),bot_bot_set_complex) ) ).

% Collect_conv_if2
tff(fact_6490_Collect__conv__if2,axiom,
    ! [A: list_nat,P2: fun_list_nat_o] :
      ( collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_hy(A),P2)) = $ite(aa_list_nat_o(P2,A),insert_list_nat(A,bot_bot_set_list_nat),bot_bot_set_list_nat) ) ).

% Collect_conv_if2
tff(fact_6491_Collect__conv__if2,axiom,
    ! [A: set_nat,P2: fun_set_nat_o] :
      ( collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_hz(A),P2)) = $ite(aa_set_nat_o(P2,A),aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat),bot_bot_set_set_nat) ) ).

% Collect_conv_if2
tff(fact_6492_Collect__conv__if2,axiom,
    ! [A: nat,P2: fun_nat_o] :
      ( collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ia(A),P2)) = $ite(aa_nat_o(P2,A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat),bot_bot_set_nat) ) ).

% Collect_conv_if2
tff(fact_6493_Collect__conv__if2,axiom,
    ! [A: int,P2: fun_int_o] :
      ( collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ib(A),P2)) = $ite(aa_int_o(P2,A),aa_set_int_set_int(insert_int(A),bot_bot_set_int),bot_bot_set_int) ) ).

% Collect_conv_if2
tff(fact_6494_Collect__conv__if2,axiom,
    ! [A: $o,P2: fun_o_o] :
      ( collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ic((A)),P2)) = $ite(aa_o_o(P2,(A)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o),bot_bot_set_o) ) ).

% Collect_conv_if2
tff(fact_6495_Collect__conv__if,axiom,
    ! [A: vEBT_VEBT,P2: fun_VEBT_VEBT_o] :
      ( collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_id(A),P2)) = $ite(aa_VEBT_VEBT_o(P2,A),aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ) ).

% Collect_conv_if
tff(fact_6496_Collect__conv__if,axiom,
    ! [A: complex,P2: fun_complex_o] :
      ( collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ie(A),P2)) = $ite(aa_complex_o(P2,A),insert_complex(A,bot_bot_set_complex),bot_bot_set_complex) ) ).

% Collect_conv_if
tff(fact_6497_Collect__conv__if,axiom,
    ! [A: list_nat,P2: fun_list_nat_o] :
      ( collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_if(A),P2)) = $ite(aa_list_nat_o(P2,A),insert_list_nat(A,bot_bot_set_list_nat),bot_bot_set_list_nat) ) ).

% Collect_conv_if
tff(fact_6498_Collect__conv__if,axiom,
    ! [A: set_nat,P2: fun_set_nat_o] :
      ( collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ig(A),P2)) = $ite(aa_set_nat_o(P2,A),aa_set1013257548857649334et_nat(insert_set_nat(A),bot_bot_set_set_nat),bot_bot_set_set_nat) ) ).

% Collect_conv_if
tff(fact_6499_Collect__conv__if,axiom,
    ! [A: nat,P2: fun_nat_o] :
      ( collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ih(A),P2)) = $ite(aa_nat_o(P2,A),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat),bot_bot_set_nat) ) ).

% Collect_conv_if
tff(fact_6500_Collect__conv__if,axiom,
    ! [A: int,P2: fun_int_o] :
      ( collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ii(A),P2)) = $ite(aa_int_o(P2,A),aa_set_int_set_int(insert_int(A),bot_bot_set_int),bot_bot_set_int) ) ).

% Collect_conv_if
tff(fact_6501_Collect__conv__if,axiom,
    ! [A: $o,P2: fun_o_o] :
      ( collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ij((A)),P2)) = $ite(aa_o_o(P2,(A)),aa_set_o_set_o(insert_o((A)),bot_bot_set_o),bot_bot_set_o) ) ).

% Collect_conv_if
tff(fact_6502_sum_Oswap__restrict,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_nat,G: fun_VE597728521808591204at_nat,R2: fun_VE6480751511521678896_nat_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups771621172384141258BT_nat(aa_fun3621025987637128233BT_nat(aa_fun1603736613815373464BT_nat(aTP_Lamm_il(B4),G),R2),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun6249118799011721995at_nat(aa_fun2705500642357200114at_nat(aTP_Lamm_in(A4),G),R2)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6503_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_nat,G: fun_real_fun_nat_nat,R2: fun_real_fun_nat_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups1935376822645274424al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_ip(B4),G),R2),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_ir(A4),G),R2)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6504_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_nat,G: fun_int_fun_nat_nat,R2: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_it(B4),G),R2),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_iw(A4),G),R2)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6505_sum_Oswap__restrict,axiom,
    ! [A4: set_complex,B4: set_nat,G: fun_co636942507509095880at_nat,R2: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups5693394587270226106ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_iy(B4),G),R2),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_jb(A4),G),R2)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6506_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B4: set_nat,G: fun_Co3312552843764780407at_nat,R2: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups7237345082560585321er_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_jd(B4),G),R2),A4) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_jg(A4),G),R2)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6507_sum_Oswap__restrict,axiom,
    ! [A4: set_VEBT_VEBT,B4: set_complex,G: fun_VE2006399705683203104omplex,R2: fun_VE7631951351884037842plex_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups1794756597179926696omplex(aa_fun1953711888481404901omplex(aa_fun5829985485761010584omplex(aTP_Lamm_ji(B4),G),R2),A4) = groups7754918857620584856omplex(aa_fun2592672125884331625omplex(aa_fun8556514703313836300omplex(aTP_Lamm_jk(A4),G),R2),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6508_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B4: set_complex,G: fun_re7208826350674946562omplex,R2: fun_re7319081385389557872plex_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5754745047067104278omplex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_jm(B4),G),R2),A4) = groups7754918857620584856omplex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_jo(A4),G),R2),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6509_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B4: set_complex,G: fun_na1591161823130732966omplex,R2: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups2073611262835488442omplex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_jq(B4),G),R2),A4) = groups7754918857620584856omplex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_jt(A4),G),R2),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6510_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B4: set_complex,G: fun_in1307110748760623106omplex,R2: fun_in5065666251602127472plex_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups3049146728041665814omplex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_jv(B4),G),R2),A4) = groups7754918857620584856omplex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_jy(A4),G),R2),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6511_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B4: set_complex,G: fun_Co8905303033605938227omplex,R2: fun_Co8975588572093281791plex_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups8024822376189712711omplex(aa_fun9008897831946212043omplex(aa_fun8319717443755388649omplex(aTP_Lamm_ka(B4),G),R2),A4) = groups7754918857620584856omplex(aa_fun7787660480616508700omplex(aa_fun963654016877434554omplex(aTP_Lamm_kd(A4),G),R2),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6512_sum__diff__distrib,axiom,
    ! [Q2: fun_int_nat,P2: fun_int_nat,Na: int] :
      ( ! [X: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Q2,X)),aa_int_nat(P2,X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P2,set_ord_lessThan_int(Na))),groups4541462559716669496nt_nat(Q2,set_ord_lessThan_int(Na))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ke(Q2),P2),set_ord_lessThan_int(Na)) ) ) ).

% sum_diff_distrib
tff(fact_6513_sum__diff__distrib,axiom,
    ! [Q2: fun_real_nat,P2: fun_real_nat,Na: real] :
      ( ! [X: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q2,X)),aa_real_nat(P2,X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P2,set_or5984915006950818249n_real(Na))),groups1935376822645274424al_nat(Q2,set_or5984915006950818249n_real(Na))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_kf(Q2),P2),set_or5984915006950818249n_real(Na)) ) ) ).

% sum_diff_distrib
tff(fact_6514_sum__diff__distrib,axiom,
    ! [Q2: fun_nat_nat,P2: fun_nat_nat,Na: nat] :
      ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q2,X)),aa_nat_nat(P2,X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P2),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(Q2),aa_nat_set_nat(set_ord_lessThan_nat,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kg(Q2),P2)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ) ) ).

% sum_diff_distrib
tff(fact_6515_max__def__raw,axiom,
    ! [X3: code_integer,Xa2: code_integer] :
      ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(X3),Xa2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa2),Xa2,X3) ) ).

% max_def_raw
tff(fact_6516_max__def__raw,axiom,
    ! [X3: set_int,Xa2: set_int] :
      ( ord_max_set_int(X3,Xa2) = $ite(aa_set_int_o(ord_less_eq_set_int(X3),Xa2),Xa2,X3) ) ).

% max_def_raw
tff(fact_6517_max__def__raw,axiom,
    ! [X3: rat,Xa2: rat] :
      ( aa_rat_rat(ord_max_rat(X3),Xa2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa2),Xa2,X3) ) ).

% max_def_raw
tff(fact_6518_max__def__raw,axiom,
    ! [X3: num,Xa2: num] :
      ( aa_num_num(ord_max_num(X3),Xa2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa2),Xa2,X3) ) ).

% max_def_raw
tff(fact_6519_max__def__raw,axiom,
    ! [X3: nat,Xa2: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Xa2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa2),Xa2,X3) ) ).

% max_def_raw
tff(fact_6520_max__def__raw,axiom,
    ! [X3: int,Xa2: int] :
      ( aa_int_int(ord_max_int(X3),Xa2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa2),Xa2,X3) ) ).

% max_def_raw
tff(fact_6521_pred__subset__eq2,axiom,
    ! [R2: set_Pr1773385645901665561uint32,S2: set_Pr1773385645901665561uint32] :
      ( ord_le3760176672600609426nt32_o(aTP_Lamm_kh(R2),aTP_Lamm_kh(S2))
    <=> ord_le6429528607962791097uint32(R2,S2) ) ).

% pred_subset_eq2
tff(fact_6522_pred__subset__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_ki(R2),aTP_Lamm_ki(S2))
    <=> ord_le3146513528884898305at_nat(R2,S2) ) ).

% pred_subset_eq2
tff(fact_6523_pred__subset__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_kj(R2),aTP_Lamm_kj(S2))
    <=> ord_le2843351958646193337nt_int(R2,S2) ) ).

% pred_subset_eq2
tff(fact_6524_pred__subset__eq2,axiom,
    ! [R2: set_Pr1312406793078187350et_nat,S2: set_Pr1312406793078187350et_nat] :
      ( ord_le3580378406483493347_nat_o(aTP_Lamm_kk(R2),aTP_Lamm_kk(S2))
    <=> ord_le3919382008627836342et_nat(R2,S2) ) ).

% pred_subset_eq2
tff(fact_6525_pred__subset__eq2,axiom,
    ! [R2: set_Pr4040394263813967220et_nat,S2: set_Pr4040394263813967220et_nat] :
      ( ord_le2834457679758710341_nat_o(aTP_Lamm_kl(R2),aTP_Lamm_kl(S2))
    <=> ord_le5189643068899110356et_nat(R2,S2) ) ).

% pred_subset_eq2
tff(fact_6526_bot__empty__eq2,axiom,
    ! [X3: uint32,Xa2: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(bot_bo3105256874341850974nt32_o,X3),Xa2)
    <=> member8027108493173000802uint32(produc1400373151660368625uint32(X3,Xa2),bot_bo8438649754162204037uint32) ) ).

% bot_empty_eq2
tff(fact_6527_bot__empty__eq2,axiom,
    ! [X3: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X3),Xa2)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa2),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_6528_bot__empty__eq2,axiom,
    ! [X3: int,Xa2: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X3),Xa2)
    <=> member5262025264175285858nt_int(product_Pair_int_int(X3,Xa2),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_6529_bot__empty__eq2,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o,Xa2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(bot_bo3509460380234399255_nat_o,X3),Xa2)
    <=> member4513763875858039863et_nat(produc6017135503889078682et_nat(X3,Xa2),bot_bo1493351289776401898et_nat) ) ).

% bot_empty_eq2
tff(fact_6530_bot__empty__eq2,axiom,
    ! [X3: fun_Pr6460944699146269935_nat_o,Xa2: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(bot_bo5031374944739956857_nat_o,X3),Xa2)
    <=> member9174161442226468181et_nat(produc7291855603632349880et_nat(X3,Xa2),bot_bo6121529163330120712et_nat) ) ).

% bot_empty_eq2
tff(fact_6531_finite__M__bounded__by__nat,axiom,
    ! [P2: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_km(P2),I))) ).

% finite_M_bounded_by_nat
tff(fact_6532_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F,N))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_kn(F),U))) ) ).

% finite_less_ub
tff(fact_6533_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : ( vEBT_set_vebt(T) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T)) ) ).

% set_vebt_def
tff(fact_6534_zip__inj,axiom,
    ! [A: list_VEBT_VEBT,B: list_VEBT_VEBT,A2: list_VEBT_VEBT,B2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(A) = size_s6755466524823107622T_VEBT(B) )
     => ( ( size_s6755466524823107622T_VEBT(A2) = size_s6755466524823107622T_VEBT(B2) )
       => ( ( zip_VE537291747668921783T_VEBT(A,B) = zip_VE537291747668921783T_VEBT(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6535_zip__inj,axiom,
    ! [A: list_VEBT_VEBT,B: list_real,A2: list_VEBT_VEBT,B2: list_real] :
      ( ( size_s6755466524823107622T_VEBT(A) = size_size_list_real(B) )
     => ( ( size_s6755466524823107622T_VEBT(A2) = size_size_list_real(B2) )
       => ( ( zip_VEBT_VEBT_real(A,B) = zip_VEBT_VEBT_real(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6536_zip__inj,axiom,
    ! [A: list_VEBT_VEBT,B: list_o,A2: list_VEBT_VEBT,B2: list_o] :
      ( ( size_s6755466524823107622T_VEBT(A) = size_size_list_o(B) )
     => ( ( size_s6755466524823107622T_VEBT(A2) = size_size_list_o(B2) )
       => ( ( zip_VEBT_VEBT_o(A,B) = zip_VEBT_VEBT_o(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6537_zip__inj,axiom,
    ! [A: list_VEBT_VEBT,B: list_nat,A2: list_VEBT_VEBT,B2: list_nat] :
      ( ( size_s6755466524823107622T_VEBT(A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( size_s6755466524823107622T_VEBT(A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_VEBT_VEBT_nat(A,B) = zip_VEBT_VEBT_nat(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6538_zip__inj,axiom,
    ! [A: list_real,B: list_VEBT_VEBT,A2: list_real,B2: list_VEBT_VEBT] :
      ( ( size_size_list_real(A) = size_s6755466524823107622T_VEBT(B) )
     => ( ( size_size_list_real(A2) = size_s6755466524823107622T_VEBT(B2) )
       => ( ( zip_real_VEBT_VEBT(A,B) = zip_real_VEBT_VEBT(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6539_zip__inj,axiom,
    ! [A: list_real,B: list_real,A2: list_real,B2: list_real] :
      ( ( size_size_list_real(A) = size_size_list_real(B) )
     => ( ( size_size_list_real(A2) = size_size_list_real(B2) )
       => ( ( zip_real_real(A,B) = zip_real_real(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6540_zip__inj,axiom,
    ! [A: list_real,B: list_o,A2: list_real,B2: list_o] :
      ( ( size_size_list_real(A) = size_size_list_o(B) )
     => ( ( size_size_list_real(A2) = size_size_list_o(B2) )
       => ( ( zip_real_o(A,B) = zip_real_o(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6541_zip__inj,axiom,
    ! [A: list_real,B: list_nat,A2: list_real,B2: list_nat] :
      ( ( size_size_list_real(A) = aa_list_nat_nat(size_size_list_nat,B) )
     => ( ( size_size_list_real(A2) = aa_list_nat_nat(size_size_list_nat,B2) )
       => ( ( zip_real_nat(A,B) = zip_real_nat(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6542_zip__inj,axiom,
    ! [A: list_o,B: list_VEBT_VEBT,A2: list_o,B2: list_VEBT_VEBT] :
      ( ( size_size_list_o(A) = size_s6755466524823107622T_VEBT(B) )
     => ( ( size_size_list_o(A2) = size_s6755466524823107622T_VEBT(B2) )
       => ( ( zip_o_VEBT_VEBT(A,B) = zip_o_VEBT_VEBT(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6543_zip__inj,axiom,
    ! [A: list_o,B: list_real,A2: list_o,B2: list_real] :
      ( ( size_size_list_o(A) = size_size_list_real(B) )
     => ( ( size_size_list_o(A2) = size_size_list_real(B2) )
       => ( ( zip_o_real(A,B) = zip_o_real(A2,B2) )
         => ( ( A = A2 )
            & ( B = B2 ) ) ) ) ) ).

% zip_inj
tff(fact_6544_pair__list__split,axiom,
    ! [L: list_P7413028617227757229T_VEBT] :
      ~ ! [L1: list_VEBT_VEBT,L22: list_VEBT_VEBT] :
          ( ( L = zip_VE537291747668921783T_VEBT(L1,L22) )
         => ( ( size_s6755466524823107622T_VEBT(L1) = size_s6755466524823107622T_VEBT(L22) )
           => ( size_s7466405169056248089T_VEBT(L) != size_s6755466524823107622T_VEBT(L22) ) ) ) ).

% pair_list_split
tff(fact_6545_pair__list__split,axiom,
    ! [L: list_P2623026923184700063T_real] :
      ~ ! [L1: list_VEBT_VEBT,L22: list_real] :
          ( ( L = zip_VEBT_VEBT_real(L1,L22) )
         => ( ( size_s6755466524823107622T_VEBT(L1) = size_size_list_real(L22) )
           => ( size_s5035110155006384947T_real(L) != size_size_list_real(L22) ) ) ) ).

% pair_list_split
tff(fact_6546_pair__list__split,axiom,
    ! [L: list_P3126845725202233233VEBT_o] :
      ~ ! [L1: list_VEBT_VEBT,L22: list_o] :
          ( ( L = zip_VEBT_VEBT_o(L1,L22) )
         => ( ( size_s6755466524823107622T_VEBT(L1) = size_size_list_o(L22) )
           => ( size_s9168528473962070013VEBT_o(L) != size_size_list_o(L22) ) ) ) ).

% pair_list_split
tff(fact_6547_pair__list__split,axiom,
    ! [L: list_P7037539587688870467BT_nat] :
      ~ ! [L1: list_VEBT_VEBT,L22: list_nat] :
          ( ( L = zip_VEBT_VEBT_nat(L1,L22) )
         => ( ( size_s6755466524823107622T_VEBT(L1) = aa_list_nat_nat(size_size_list_nat,L22) )
           => ( size_s6152045936467909847BT_nat(L) != aa_list_nat_nat(size_size_list_nat,L22) ) ) ) ).

% pair_list_split
tff(fact_6548_pair__list__split,axiom,
    ! [L: list_P877281246627933069T_VEBT] :
      ~ ! [L1: list_real,L22: list_VEBT_VEBT] :
          ( ( L = zip_real_VEBT_VEBT(L1,L22) )
         => ( ( size_size_list_real(L1) = size_s6755466524823107622T_VEBT(L22) )
           => ( size_s3289364478449617953T_VEBT(L) != size_s6755466524823107622T_VEBT(L22) ) ) ) ).

% pair_list_split
tff(fact_6549_pair__list__split,axiom,
    ! [L: list_P8689742595348180415l_real] :
      ~ ! [L1: list_real,L22: list_real] :
          ( ( L = zip_real_real(L1,L22) )
         => ( ( size_size_list_real(L1) = size_size_list_real(L22) )
           => ( size_s3932428310213730859l_real(L) != size_size_list_real(L22) ) ) ) ).

% pair_list_split
tff(fact_6550_pair__list__split,axiom,
    ! [L: list_P3595434254542482545real_o] :
      ~ ! [L1: list_real,L22: list_o] :
          ( ( L = zip_real_o(L1,L22) )
         => ( ( size_size_list_real(L1) = size_size_list_o(L22) )
           => ( size_s987546567493390085real_o(L) != size_size_list_o(L22) ) ) ) ).

% pair_list_split
tff(fact_6551_pair__list__split,axiom,
    ! [L: list_P6834414599653733731al_nat] :
      ~ ! [L1: list_real,L22: list_nat] :
          ( ( L = zip_real_nat(L1,L22) )
         => ( ( size_size_list_real(L1) = aa_list_nat_nat(size_size_list_nat,L22) )
           => ( size_s1877336372972134351al_nat(L) != aa_list_nat_nat(size_size_list_nat,L22) ) ) ) ).

% pair_list_split
tff(fact_6552_pair__list__split,axiom,
    ! [L: list_P7495141550334521929T_VEBT] :
      ~ ! [L1: list_o,L22: list_VEBT_VEBT] :
          ( ( L = zip_o_VEBT_VEBT(L1,L22) )
         => ( ( size_size_list_o(L1) = size_s6755466524823107622T_VEBT(L22) )
           => ( size_s4313452262239582901T_VEBT(L) != size_s6755466524823107622T_VEBT(L22) ) ) ) ).

% pair_list_split
tff(fact_6553_pair__list__split,axiom,
    ! [L: list_P5232166724548748803o_real] :
      ~ ! [L1: list_o,L22: list_real] :
          ( ( L = zip_o_real(L1,L22) )
         => ( ( size_size_list_o(L1) = size_size_list_real(L22) )
           => ( size_s2624279037499656343o_real(L) != size_size_list_real(L22) ) ) ) ).

% pair_list_split
tff(fact_6554_real__arch__simple,axiom,
    ! [Xa: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_6555_real__arch__simple,axiom,
    ! [Xa: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_6556_reals__Archimedean2,axiom,
    ! [Xa: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_6557_reals__Archimedean2,axiom,
    ! [Xa: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_6558_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: uint32] : ( aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,Xa)),Y) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,Y),aa_nat_uint32(semiri2565882477558803405uint32,Xa)) ) ).

% mult_of_nat_commute
tff(fact_6559_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xa)) ) ).

% mult_of_nat_commute
tff(fact_6560_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: real] : ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,Xa)) ) ).

% mult_of_nat_commute
tff(fact_6561_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: int] : ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% mult_of_nat_commute
tff(fact_6562_mult__of__nat__commute,axiom,
    ! [Xa: nat,Y: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)) ) ).

% mult_of_nat_commute
tff(fact_6563_lessThan__non__empty,axiom,
    ! [Xa: int] : ( set_ord_lessThan_int(Xa) != bot_bot_set_int ) ).

% lessThan_non_empty
tff(fact_6564_lessThan__non__empty,axiom,
    ! [Xa: real] : ( set_or5984915006950818249n_real(Xa) != bot_bot_set_real ) ).

% lessThan_non_empty
tff(fact_6565_infinite__Iio,axiom,
    ! [A: int] : ~ aa_set_int_o(finite_finite_int,set_ord_lessThan_int(A)) ).

% infinite_Iio
tff(fact_6566_infinite__Iio,axiom,
    ! [A: real] : ~ aa_set_real_o(finite_finite_real,set_or5984915006950818249n_real(A)) ).

% infinite_Iio
tff(fact_6567_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ko(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_6568_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_kp(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_6569_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kq(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_6570_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_kr(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_6571_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_uint32,M: nat] : ( groups833757482993574392uint32(aTP_Lamm_ks(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_uint32(aa_nat_uint32(F,M),aa_nat_uint32(F,zero_zero_nat)) ) ).

% sum_lessThan_telescope
tff(fact_6572_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_kt(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_6573_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : ( groups6591440286371151544t_real(aTP_Lamm_ku(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(minus_minus_real(aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ) ).

% sum_lessThan_telescope
tff(fact_6574_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_uint32,M: nat] : ( groups833757482993574392uint32(aTP_Lamm_kv(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_uint32(aa_nat_uint32(F,zero_zero_nat),aa_nat_uint32(F,M)) ) ).

% sum_lessThan_telescope'
tff(fact_6575_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_kw(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_6576_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : ( groups6591440286371151544t_real(aTP_Lamm_kx(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ) ).

% sum_lessThan_telescope'
tff(fact_6577_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xa: fun_VEBT_VEBT_uint32,Y: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun6272599145770227614VEBT_o(aTP_Lamm_ky(I5),Xa)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun6272599145770227614VEBT_o(aTP_Lamm_ky(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun6272599145770227614VEBT_o(aa_fun1156574460560677430VEBT_o(aTP_Lamm_kz(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6578_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_uint32,Y: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7556296611544133298real_o(aTP_Lamm_la(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7556296611544133298real_o(aTP_Lamm_la(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun7556296611544133298real_o(aa_fun5572330763077102134real_o(aTP_Lamm_lb(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6579_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_uint32,Y: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1182053767699555178_nat_o(aTP_Lamm_lc(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1182053767699555178_nat_o(aTP_Lamm_lc(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1182053767699555178_nat_o(aa_fun6957237400125433034_nat_o(aTP_Lamm_ld(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6580_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_uint32,Y: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6992502331602369202_int_o(aTP_Lamm_le(I5),Xa)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6992502331602369202_int_o(aTP_Lamm_le(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun6992502331602369202_int_o(aa_fun3369613587901038774_int_o(aTP_Lamm_lf(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6581_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xa: fun_complex_uint32,Y: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun436548534304416686plex_o(aTP_Lamm_lg(I5),Xa)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun436548534304416686plex_o(aTP_Lamm_lg(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun436548534304416686plex_o(aa_fun2751405167760845232plex_o(aTP_Lamm_lh(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6582_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,Xa: fun_Co1554742298434871494uint32,Y: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8926496376162642256eger_o(aTP_Lamm_li(I5),Xa)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8926496376162642256eger_o(aTP_Lamm_li(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8926496376162642256eger_o(aa_fun1119956648780089443eger_o(aTP_Lamm_lj(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6583_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xa: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_lk(I5),Xa)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_lk(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_ll(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6584_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_lm(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_lm(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ln(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6585_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_lo(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_lo(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_lp(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6586_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_lq(I5),Xa)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_lq(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_lr(I5),Xa),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_6587_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xa: fun_VEBT_VEBT_rat,Y: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aTP_Lamm_ls(I5),Xa)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aTP_Lamm_ls(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aa_fun1978406156034293430VEBT_o(aTP_Lamm_lt(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6588_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_rat,Y: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_lu(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_lu(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_lv(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6589_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_lw(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_lw(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_lx(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6590_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_rat,Y: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ly(I5),Xa)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ly(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_lz(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6591_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xa: fun_complex_rat,Y: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aTP_Lamm_ma(I5),Xa)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aTP_Lamm_ma(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aa_fun4586712283274990936plex_o(aTP_Lamm_mb(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6592_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,Xa: fun_Code_integer_rat,Y: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_mc(I5),Xa)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aTP_Lamm_mc(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun6395433285121891940eger_o(aa_fun4095556841599367307eger_o(aTP_Lamm_md(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6593_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xa: fun_VEBT_VEBT_assn,Y: fun_VEBT_VEBT_assn] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5142337294461892954VEBT_o(aTP_Lamm_me(I5),Xa)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5142337294461892954VEBT_o(aTP_Lamm_me(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5142337294461892954VEBT_o(aa_fun876682331325704374VEBT_o(aTP_Lamm_mf(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6594_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xa: fun_real_assn,Y: fun_real_assn] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7580339382299851254real_o(aTP_Lamm_mg(I5),Xa)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7580339382299851254real_o(aTP_Lamm_mg(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun7580339382299851254real_o(aa_fun4959323011771448382real_o(aTP_Lamm_mh(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6595_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xa: fun_nat_assn,Y: fun_nat_assn] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aTP_Lamm_mi(I5),Xa)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aTP_Lamm_mi(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun7183233715851567790_nat_o(aa_fun8595640264602358482_nat_o(aTP_Lamm_mj(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6596_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xa: fun_int_assn,Y: fun_int_assn] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aTP_Lamm_mk(I5),Xa)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aTP_Lamm_mk(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun202081646129719286_int_o(aa_fun871127997271750846_int_o(aTP_Lamm_ml(I5),Xa),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_6597_filter__preserves__multiset,axiom,
    ! [M7: fun_list_nat_nat,P2: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_mm(M7)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_mn(M7),P2))) ) ).

% filter_preserves_multiset
tff(fact_6598_filter__preserves__multiset,axiom,
    ! [M7: fun_set_nat_nat,P2: fun_set_nat_o] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_mo(M7)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_mp(M7),P2))) ) ).

% filter_preserves_multiset
tff(fact_6599_filter__preserves__multiset,axiom,
    ! [M7: fun_nat_nat,P2: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_mq(M7)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_mr(M7),P2))) ) ).

% filter_preserves_multiset
tff(fact_6600_filter__preserves__multiset,axiom,
    ! [M7: fun_int_nat,P2: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_ms(M7)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_mt(M7),P2))) ) ).

% filter_preserves_multiset
tff(fact_6601_filter__preserves__multiset,axiom,
    ! [M7: fun_complex_nat,P2: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_mu(M7)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_mv(M7),P2))) ) ).

% filter_preserves_multiset
tff(fact_6602_filter__preserves__multiset,axiom,
    ! [M7: fun_Code_integer_nat,P2: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_mw(M7)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_mx(M7),P2))) ) ).

% filter_preserves_multiset
tff(fact_6603_sum_Ointer__filter,axiom,
    ! [A4: set_VEBT_VEBT,G: fun_VEBT_VEBT_uint32,P2: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups8325533452322294502uint32(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_dt(A4),P2))) = groups8325533452322294502uint32(aa_fun2089568865546376974uint32(aTP_Lamm_my(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6604_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_uint32,P2: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups5944083974425963860uint32(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_du(A4),P2))) = groups5944083974425963860uint32(aa_fun7747623534341301918uint32(aTP_Lamm_mz(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6605_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G: fun_nat_uint32,P2: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( groups833757482993574392uint32(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_dy(A4),P2))) = groups833757482993574392uint32(aa_fun7328656920723993502uint32(aTP_Lamm_na(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6606_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_uint32,P2: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups5712668689793887828uint32(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dz(A4),P2))) = groups5712668689793887828uint32(aa_fun1392314963150257310uint32(aTP_Lamm_nb(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6607_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_uint32,P2: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups8736914816313324502uint32(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dv(A4),P2))) = groups8736914816313324502uint32(aa_fun9209306604415416350uint32(aTP_Lamm_nc(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6608_sum_Ointer__filter,axiom,
    ! [A4: set_Code_integer,G: fun_Co1554742298434871494uint32,P2: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups8847630953604152069uint32(G,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_nd(A4),P2))) = groups8847630953604152069uint32(aa_fun7787524580160642142uint32(aTP_Lamm_ne(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6609_sum_Ointer__filter,axiom,
    ! [A4: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P2: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( groups2240296850493347238T_real(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_dt(A4),P2))) = groups2240296850493347238T_real(aa_fun3597767889293385422T_real(aTP_Lamm_nf(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6610_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: fun_real_real,P2: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_du(A4),P2))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_ng(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6611_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: fun_int_real,P2: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_dz(A4),P2))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_nh(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6612_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: fun_complex_real,P2: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_dv(A4),P2))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_ni(G),P2),A4) ) ) ).

% sum.inter_filter
tff(fact_6613_image__constant,axiom,
    ! [Xa: nat,A4: set_nat,C2: vEBT_VEBT] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( image_nat_VEBT_VEBT(aTP_Lamm_nj(C2),A4) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C2),bot_bo8194388402131092736T_VEBT) ) ) ).

% image_constant
tff(fact_6614_image__constant,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( image_3375948659692109573T_VEBT(aTP_Lamm_nk(C2),A4) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C2),bot_bo8194388402131092736T_VEBT) ) ) ).

% image_constant
tff(fact_6615_image__constant,axiom,
    ! [Xa: real,A4: set_real,C2: vEBT_VEBT] :
      ( aa_set_real_o(member_real(Xa),A4)
     => ( image_real_VEBT_VEBT(aTP_Lamm_nl(C2),A4) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C2),bot_bo8194388402131092736T_VEBT) ) ) ).

% image_constant
tff(fact_6616_image__constant,axiom,
    ! [Xa: int,A4: set_int,C2: vEBT_VEBT] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( image_int_VEBT_VEBT(aTP_Lamm_nm(C2),A4) = aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C2),bot_bo8194388402131092736T_VEBT) ) ) ).

% image_constant
tff(fact_6617_image__constant,axiom,
    ! [Xa: nat,A4: set_nat,C2: nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_cz(C2)),A4) = aa_set_nat_set_nat(insert_nat(C2),bot_bot_set_nat) ) ) ).

% image_constant
tff(fact_6618_image__constant,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,C2: nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( image_VEBT_VEBT_nat(aTP_Lamm_nn(C2),A4) = aa_set_nat_set_nat(insert_nat(C2),bot_bot_set_nat) ) ) ).

% image_constant
tff(fact_6619_image__constant,axiom,
    ! [Xa: real,A4: set_real,C2: nat] :
      ( aa_set_real_o(member_real(Xa),A4)
     => ( image_real_nat(aTP_Lamm_no(C2),A4) = aa_set_nat_set_nat(insert_nat(C2),bot_bot_set_nat) ) ) ).

% image_constant
tff(fact_6620_image__constant,axiom,
    ! [Xa: int,A4: set_int,C2: nat] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( aa_set_int_set_nat(image_int_nat(aTP_Lamm_da(C2)),A4) = aa_set_nat_set_nat(insert_nat(C2),bot_bot_set_nat) ) ) ).

% image_constant
tff(fact_6621_image__constant,axiom,
    ! [Xa: nat,A4: set_nat,C2: int] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_set_int(image_nat_int(aTP_Lamm_cv(C2)),A4) = aa_set_int_set_int(insert_int(C2),bot_bot_set_int) ) ) ).

% image_constant
tff(fact_6622_image__constant,axiom,
    ! [Xa: vEBT_VEBT,A4: set_VEBT_VEBT,C2: int] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),A4)
     => ( image_VEBT_VEBT_int(aTP_Lamm_np(C2),A4) = aa_set_int_set_int(insert_int(C2),bot_bot_set_int) ) ) ).

% image_constant
tff(fact_6623_image__constant__conv,axiom,
    ! [C2: vEBT_VEBT,A4: set_nat] :
      ( image_nat_VEBT_VEBT(aTP_Lamm_nj(C2),A4) = $ite(A4 = bot_bot_set_nat,bot_bo8194388402131092736T_VEBT,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C2),bot_bo8194388402131092736T_VEBT)) ) ).

% image_constant_conv
tff(fact_6624_image__constant__conv,axiom,
    ! [C2: nat,A4: set_nat] :
      ( aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_cz(C2)),A4) = $ite(A4 = bot_bot_set_nat,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(C2),bot_bot_set_nat)) ) ).

% image_constant_conv
tff(fact_6625_image__constant__conv,axiom,
    ! [C2: int,A4: set_nat] :
      ( aa_set_nat_set_int(image_nat_int(aTP_Lamm_cv(C2)),A4) = $ite(A4 = bot_bot_set_nat,bot_bot_set_int,aa_set_int_set_int(insert_int(C2),bot_bot_set_int)) ) ).

% image_constant_conv
tff(fact_6626_image__constant__conv,axiom,
    ! [C2: $o,A4: set_nat] :
      ( image_nat_o(aa_o_fun_nat_o(aTP_Lamm_nq,(C2)),A4) = $ite(A4 = bot_bot_set_nat,bot_bot_set_o,aa_set_o_set_o(insert_o((C2)),bot_bot_set_o)) ) ).

% image_constant_conv
tff(fact_6627_image__constant__conv,axiom,
    ! [C2: vEBT_VEBT,A4: set_int] :
      ( image_int_VEBT_VEBT(aTP_Lamm_nm(C2),A4) = $ite(A4 = bot_bot_set_int,bot_bo8194388402131092736T_VEBT,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C2),bot_bo8194388402131092736T_VEBT)) ) ).

% image_constant_conv
tff(fact_6628_image__constant__conv,axiom,
    ! [C2: nat,A4: set_int] :
      ( aa_set_int_set_nat(image_int_nat(aTP_Lamm_da(C2)),A4) = $ite(A4 = bot_bot_set_int,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(C2),bot_bot_set_nat)) ) ).

% image_constant_conv
tff(fact_6629_image__constant__conv,axiom,
    ! [C2: int,A4: set_int] :
      ( aa_set_int_set_int(image_int_int(aTP_Lamm_cw(C2)),A4) = $ite(A4 = bot_bot_set_int,bot_bot_set_int,aa_set_int_set_int(insert_int(C2),bot_bot_set_int)) ) ).

% image_constant_conv
tff(fact_6630_image__constant__conv,axiom,
    ! [C2: $o,A4: set_int] :
      ( image_int_o(aTP_Lamm_nr((C2)),A4) = $ite(A4 = bot_bot_set_int,bot_bot_set_o,aa_set_o_set_o(insert_o((C2)),bot_bot_set_o)) ) ).

% image_constant_conv
tff(fact_6631_image__constant__conv,axiom,
    ! [C2: vEBT_VEBT,A4: set_o] :
      ( image_o_VEBT_VEBT(aTP_Lamm_ns(C2),A4) = $ite(A4 = bot_bot_set_o,bot_bo8194388402131092736T_VEBT,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(C2),bot_bo8194388402131092736T_VEBT)) ) ).

% image_constant_conv
tff(fact_6632_image__constant__conv,axiom,
    ! [C2: nat,A4: set_o] :
      ( image_o_nat(aTP_Lamm_db(C2),A4) = $ite(A4 = bot_bot_set_o,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(C2),bot_bot_set_nat)) ) ).

% image_constant_conv
tff(fact_6633_foldl__length__aux,axiom,
    ! [A: nat,L: list_VEBT_VEBT] : ( foldl_nat_VEBT_VEBT(aTP_Lamm_cn,A,L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_s6755466524823107622T_VEBT(L)) ) ).

% foldl_length_aux
tff(fact_6634_foldl__length__aux,axiom,
    ! [A: nat,L: list_real] : ( foldl_nat_real(aTP_Lamm_co,A,L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_real(L)) ) ).

% foldl_length_aux
tff(fact_6635_foldl__length__aux,axiom,
    ! [A: nat,L: list_o] : ( foldl_nat_o(aTP_Lamm_cp,A,L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_o(L)) ) ).

% foldl_length_aux
tff(fact_6636_foldl__length__aux,axiom,
    ! [A: nat,L: list_nat] : ( aa_list_nat_nat(foldl_nat_nat(aTP_Lamm_cq,A),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_nat_nat(size_size_list_nat,L)) ) ).

% foldl_length_aux
tff(fact_6637_sum_Oimage__gen,axiom,
    ! [S2: set_VEBT_VEBT,H2: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups771621172384141258BT_nat(H2,S2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun4135997101171927032at_nat(aa_fun456795606997707180at_nat(aTP_Lamm_nu(S2),H2),G)),image_VEBT_VEBT_nat(G,S2)) ) ) ).

% sum.image_gen
tff(fact_6638_sum_Oimage__gen,axiom,
    ! [S2: set_real,H2: fun_real_nat,G: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1935376822645274424al_nat(H2,S2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3947338165632262996at_nat(aa_fun3815547947404115146at_nat(aTP_Lamm_nw(S2),H2),G)),image_real_nat(G,S2)) ) ) ).

% sum.image_gen
tff(fact_6639_sum_Oimage__gen,axiom,
    ! [S2: set_int,H2: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups4541462559716669496nt_nat(H2,S2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_ny(S2),H2),G)),aa_set_int_set_nat(image_int_nat(G),S2)) ) ) ).

% sum.image_gen
tff(fact_6640_sum_Oimage__gen,axiom,
    ! [S2: set_complex,H2: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5693394587270226106ex_nat(H2,S2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3611238026331006550at_nat(aa_fun4158405052476944462at_nat(aTP_Lamm_oa(S2),H2),G)),image_complex_nat(G,S2)) ) ) ).

% sum.image_gen
tff(fact_6641_sum_Oimage__gen,axiom,
    ! [S2: set_Code_integer,H2: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups7237345082560585321er_nat(H2,S2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_oc(S2),H2),G)),image_951025933927791156er_nat(G,S2)) ) ) ).

% sum.image_gen
tff(fact_6642_sum_Oimage__gen,axiom,
    ! [S2: set_VEBT_VEBT,H2: fun_VE5569070535936029429omplex,G: fun_VE5569070535936029429omplex] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups1794756597179926696omplex(H2,S2) = groups7754918857620584856omplex(aa_fun7854130243116081298omplex(aa_fun1272854850243878376omplex(aTP_Lamm_oe(S2),H2),G),image_3793382806556112285omplex(G,S2)) ) ) ).

% sum.image_gen
tff(fact_6643_sum_Oimage__gen,axiom,
    ! [S2: set_real,H2: fun_real_complex,G: fun_real_complex] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups5754745047067104278omplex(H2,S2) = groups7754918857620584856omplex(aa_fun5897791145500046318omplex(aa_fun9191395874192406082omplex(aTP_Lamm_og(S2),H2),G),image_real_complex(G,S2)) ) ) ).

% sum.image_gen
tff(fact_6644_sum_Oimage__gen,axiom,
    ! [S2: set_nat,H2: fun_nat_complex,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2073611262835488442omplex(H2,S2) = groups7754918857620584856omplex(aa_fun711518050430204434omplex(aa_fun2227483609818284298omplex(aTP_Lamm_oi(S2),H2),G),image_nat_complex(G,S2)) ) ) ).

% sum.image_gen
tff(fact_6645_sum_Oimage__gen,axiom,
    ! [S2: set_int,H2: fun_int_complex,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3049146728041665814omplex(H2,S2) = groups7754918857620584856omplex(aa_fun2120204959111876718omplex(aa_fun1822015382586975682omplex(aTP_Lamm_ok(S2),H2),G),image_int_complex(G,S2)) ) ) ).

% sum.image_gen
tff(fact_6646_sum_Oimage__gen,axiom,
    ! [S2: set_Code_integer,H2: fun_Co1900716531650368264omplex,G: fun_Co1900716531650368264omplex] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups8024822376189712711omplex(H2,S2) = groups7754918857620584856omplex(aa_fun4919649359853996703omplex(aa_fun7424176652353795108omplex(aTP_Lamm_om(S2),H2),G),image_3397630267976458002omplex(G,S2)) ) ) ).

% sum.image_gen
tff(fact_6647_nat__seg__image__imp__finite,axiom,
    ! [A4: set_set_nat,F: fun_nat_set_nat,Na: nat] :
      ( ( A4 = aa_set7246812951530217472et_nat(image_nat_set_nat(F),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,Na))) )
     => aa_set_set_nat_o(finite1152437895449049373et_nat,A4) ) ).

% nat_seg_image_imp_finite
tff(fact_6648_nat__seg__image__imp__finite,axiom,
    ! [A4: set_nat,F: fun_nat_nat,Na: nat] :
      ( ( A4 = aa_set_nat_set_nat(image_nat_nat(F),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,Na))) )
     => aa_set_nat_o(finite_finite_nat,A4) ) ).

% nat_seg_image_imp_finite
tff(fact_6649_nat__seg__image__imp__finite,axiom,
    ! [A4: set_int,F: fun_nat_int,Na: nat] :
      ( ( A4 = aa_set_nat_set_int(image_nat_int(F),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,Na))) )
     => aa_set_int_o(finite_finite_int,A4) ) ).

% nat_seg_image_imp_finite
tff(fact_6650_nat__seg__image__imp__finite,axiom,
    ! [A4: set_complex,F: fun_nat_complex,Na: nat] :
      ( ( A4 = image_nat_complex(F,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,Na))) )
     => aa_set_complex_o(finite3207457112153483333omplex,A4) ) ).

% nat_seg_image_imp_finite
tff(fact_6651_nat__seg__image__imp__finite,axiom,
    ! [A4: set_Code_integer,F: fun_nat_Code_integer,Na: nat] :
      ( ( A4 = image_1215581382706833972nteger(F,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,Na))) )
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4) ) ).

% nat_seg_image_imp_finite
tff(fact_6652_finite__conv__nat__seg__image,axiom,
    ! [A4: set_set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
    <=> ? [N5: nat,F5: fun_nat_set_nat] : ( A4 = aa_set7246812951530217472et_nat(image_nat_set_nat(F5),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,N5))) ) ) ).

% finite_conv_nat_seg_image
tff(fact_6653_finite__conv__nat__seg__image,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
    <=> ? [N5: nat,F5: fun_nat_nat] : ( A4 = aa_set_nat_set_nat(image_nat_nat(F5),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,N5))) ) ) ).

% finite_conv_nat_seg_image
tff(fact_6654_finite__conv__nat__seg__image,axiom,
    ! [A4: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
    <=> ? [N5: nat,F5: fun_nat_int] : ( A4 = aa_set_nat_set_int(image_nat_int(F5),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,N5))) ) ) ).

% finite_conv_nat_seg_image
tff(fact_6655_finite__conv__nat__seg__image,axiom,
    ! [A4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
    <=> ? [N5: nat,F5: fun_nat_complex] : ( A4 = image_nat_complex(F5,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,N5))) ) ) ).

% finite_conv_nat_seg_image
tff(fact_6656_finite__conv__nat__seg__image,axiom,
    ! [A4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
    <=> ? [N5: nat,F5: fun_nat_Code_integer] : ( A4 = image_1215581382706833972nteger(F5,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,N5))) ) ) ).

% finite_conv_nat_seg_image
tff(fact_6657_finite__int__segment,axiom,
    ! [A: real,B: real] : aa_set_real_o(finite_finite_real,collect_real(aa_real_fun_real_o(aTP_Lamm_on(A),B))) ).

% finite_int_segment
tff(fact_6658_finite__int__segment,axiom,
    ! [A: rat,B: rat] : aa_set_rat_o(finite_finite_rat,collect_rat(aa_rat_fun_rat_o(aTP_Lamm_oo(A),B))) ).

% finite_int_segment
tff(fact_6659_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kq(G)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_6660_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] : ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))) = groups6591440286371151544t_real(aTP_Lamm_kr(G),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_6661_sum__subtractf__nat,axiom,
    ! [A4: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,F: fun_VEBT_VEBT_nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(G,X)),aa_VEBT_VEBT_nat(F,X)) )
     => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_op(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A4)),groups771621172384141258BT_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6662_sum__subtractf__nat,axiom,
    ! [A4: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X)),aa_real_nat(F,X)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_kf(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A4)),groups1935376822645274424al_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6663_sum__subtractf__nat,axiom,
    ! [A4: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X)),aa_int_nat(F,X)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ke(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A4)),groups4541462559716669496nt_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6664_sum__subtractf__nat,axiom,
    ! [A4: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X)),aa_set_nat_nat(F,X)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_oq(G),F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A4)),groups8294997508430121362at_nat(G,A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6665_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X)),aa_nat_nat(F,X)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kg(G),F)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_6666_sum__multicount__gen,axiom,
    ! [Sa: set_VEBT_VEBT,T: set_VEBT_VEBT,R2: fun_VE2962686792283380846VEBT_o,K: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,Sa)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),T)
             => ( finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_VEB1521802747257091359VEBT_o(aa_fun8771395773580142140VEBT_o(aTP_Lamm_or(Sa),R2),X))) = aa_VEBT_VEBT_nat(K,X) ) )
         => ( groups771621172384141258BT_nat(aa_fun2100738023942440677BT_nat(aTP_Lamm_ot(T),R2),Sa) = groups771621172384141258BT_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6667_sum__multicount__gen,axiom,
    ! [Sa: set_VEBT_VEBT,T: set_real,R2: fun_VE3823611782367660756real_o,K: fun_real_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,Sa)
     => ( aa_set_real_o(finite_finite_real,T)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),T)
             => ( finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_rea6225614514526338833VEBT_o(aa_fun575193042032787426VEBT_o(aTP_Lamm_ou(Sa),R2),X))) = aa_real_nat(K,X) ) )
         => ( groups771621172384141258BT_nat(aa_fun2935073076783396613BT_nat(aTP_Lamm_ow(T),R2),Sa) = groups1935376822645274424al_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6668_sum__multicount__gen,axiom,
    ! [Sa: set_real,T: set_VEBT_VEBT,R2: fun_re7829509182167823952VEBT_o,K: fun_VEBT_VEBT_nat] :
      ( aa_set_real_o(finite_finite_real,Sa)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),T)
             => ( finite_card_real(collect_real(aa_VEB641519613517496269real_o(aa_fun7937150527584618978real_o(aTP_Lamm_ox(Sa),R2),X))) = aa_VEBT_VEBT_nat(K,X) ) )
         => ( groups1935376822645274424al_nat(aa_fun1148221025137894919al_nat(aTP_Lamm_oz(T),R2),Sa) = groups771621172384141258BT_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6669_sum__multicount__gen,axiom,
    ! [Sa: set_real,T: set_real,R2: fun_real_fun_real_o,K: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,Sa)
     => ( aa_set_real_o(finite_finite_real,T)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),T)
             => ( finite_card_real(collect_real(aa_real_fun_real_o(aa_fun4467907209059791804real_o(aTP_Lamm_pa(Sa),R2),X))) = aa_real_nat(K,X) ) )
         => ( groups1935376822645274424al_nat(aa_fun3429493660699118127al_nat(aTP_Lamm_pc(T),R2),Sa) = groups1935376822645274424al_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6670_sum__multicount__gen,axiom,
    ! [Sa: set_VEBT_VEBT,T: set_literal,R2: fun_VE8371440389858774350eral_o,K: fun_literal_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,Sa)
     => ( finite5847741373460823677iteral(T)
       => ( ! [X: literal] :
              ( member_literal(X,T)
             => ( finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_lit1284886718724587071VEBT_o(aa_fun5379558016612791804VEBT_o(aTP_Lamm_pd(Sa),R2),X))) = aa_literal_nat(K,X) ) )
         => ( groups771621172384141258BT_nat(aa_fun3014642116068820613BT_nat(aTP_Lamm_pf(T),R2),Sa) = groups8652099787943017962al_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6671_sum__multicount__gen,axiom,
    ! [Sa: set_real,T: set_literal,R2: fun_re3850508403959675952eral_o,K: fun_literal_nat] :
      ( aa_set_real_o(finite_finite_real,Sa)
     => ( finite5847741373460823677iteral(T)
       => ( ! [X: literal] :
              ( member_literal(X,T)
             => ( finite_card_real(collect_real(aa_lit1368651074454191789real_o(aa_fun3100033197786232354real_o(aTP_Lamm_pg(Sa),R2),X))) = aa_literal_nat(K,X) ) )
         => ( groups1935376822645274424al_nat(aa_fun705237418523604071al_nat(aTP_Lamm_pi(T),R2),Sa) = groups8652099787943017962al_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6672_sum__multicount__gen,axiom,
    ! [Sa: set_literal,T: set_VEBT_VEBT,R2: fun_li8107907266037299598VEBT_o,K: fun_VEBT_VEBT_nat] :
      ( finite5847741373460823677iteral(Sa)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),T)
             => ( finite_card_literal(collect_literal(aa_VEB1719478621182999551eral_o(aa_fun1102782463918247292eral_o(aTP_Lamm_pj(Sa),R2),X))) = aa_VEBT_VEBT_nat(K,X) ) )
         => ( groups8652099787943017962al_nat(aa_fun1876825894470965861al_nat(aTP_Lamm_pl(T),R2),Sa) = groups771621172384141258BT_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6673_sum__multicount__gen,axiom,
    ! [Sa: set_literal,T: set_real,R2: fun_li5129792558484764084real_o,K: fun_real_nat] :
      ( finite5847741373460823677iteral(Sa)
     => ( aa_set_real_o(finite_finite_real,T)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),T)
             => ( finite_card_literal(collect_literal(aa_rea1838623199774720753eral_o(aa_fun5708556627802011170eral_o(aTP_Lamm_pm(Sa),R2),X))) = aa_real_nat(K,X) ) )
         => ( groups8652099787943017962al_nat(aa_fun202459941351508933al_nat(aTP_Lamm_po(T),R2),Sa) = groups1935376822645274424al_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6674_sum__multicount__gen,axiom,
    ! [Sa: set_literal,T: set_literal,R2: fun_li4177005666941401710eral_o,K: fun_literal_nat] :
      ( finite5847741373460823677iteral(Sa)
     => ( finite5847741373460823677iteral(T)
       => ( ! [X: literal] :
              ( member_literal(X,T)
             => ( finite_card_literal(collect_literal(aa_lit5986169069484365087eral_o(aa_fun6827806182750346556eral_o(aTP_Lamm_pp(Sa),R2),X))) = aa_literal_nat(K,X) ) )
         => ( groups8652099787943017962al_nat(aa_fun8110944661589842437al_nat(aTP_Lamm_pr(T),R2),Sa) = groups8652099787943017962al_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6675_sum__multicount__gen,axiom,
    ! [Sa: set_VEBT_VEBT,T: set_int,R2: fun_VE4235606543851232980_int_o,K: fun_int_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,Sa)
     => ( aa_set_int_o(finite_finite_int,T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),T)
             => ( finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_int8839263053405020433VEBT_o(aa_fun8807180201667926370VEBT_o(aTP_Lamm_fv(Sa),R2),X))) = aa_int_nat(K,X) ) )
         => ( groups771621172384141258BT_nat(aa_fun5588456476524577925BT_nat(aTP_Lamm_pt(T),R2),Sa) = groups4541462559716669496nt_nat(K,T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_6676_total__onI,axiom,
    ! [A4: set_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( ! [X: vEBT_VEBT,Y3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A4)
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y3),A4)
           => ( ( X != Y3 )
             => ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X,Y3),R)
                | member568628332442017744T_VEBT(produc537772716801021591T_VEBT(Y3,X),R) ) ) ) )
     => total_on_VEBT_VEBT(A4,R) ) ).

% total_onI
tff(fact_6677_total__onI,axiom,
    ! [A4: set_real,R: set_Pr6218003697084177305l_real] :
      ( ! [X: real,Y3: real] :
          ( aa_set_real_o(member_real(X),A4)
         => ( aa_set_real_o(member_real(Y3),A4)
           => ( ( X != Y3 )
             => ( member7849222048561428706l_real(produc4511245868158468465l_real(X,Y3),R)
                | member7849222048561428706l_real(produc4511245868158468465l_real(Y3,X),R) ) ) ) )
     => total_on_real(A4,R) ) ).

% total_onI
tff(fact_6678_total__onI,axiom,
    ! [A4: set_set_nat,R: set_Pr5488025237498180813et_nat] :
      ( ! [X: set_nat,Y3: set_nat] :
          ( aa_set_set_nat_o(member_set_nat(X),A4)
         => ( aa_set_set_nat_o(member_set_nat(Y3),A4)
           => ( ( X != Y3 )
             => ( member8277197624267554838et_nat(produc4532415448927165861et_nat(X,Y3),R)
                | member8277197624267554838et_nat(produc4532415448927165861et_nat(Y3,X),R) ) ) ) )
     => total_on_set_nat(A4,R) ) ).

% total_onI
tff(fact_6679_total__onI,axiom,
    ! [A4: set_uint32,R: set_Pr1773385645901665561uint32] :
      ( ! [X: uint32,Y3: uint32] :
          ( member_uint32(X,A4)
         => ( member_uint32(Y3,A4)
           => ( ( X != Y3 )
             => ( member8027108493173000802uint32(produc1400373151660368625uint32(X,Y3),R)
                | member8027108493173000802uint32(produc1400373151660368625uint32(Y3,X),R) ) ) ) )
     => total_on_uint32(A4,R) ) ).

% total_onI
tff(fact_6680_total__onI,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y3: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => ( aa_set_nat_o(member_nat(Y3),A4)
           => ( ( X != Y3 )
             => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3),R)
                | member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y3),X),R) ) ) ) )
     => total_on_nat(A4,R) ) ).

% total_onI
tff(fact_6681_total__onI,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int] :
      ( ! [X: int,Y3: int] :
          ( aa_set_int_o(member_int(X),A4)
         => ( aa_set_int_o(member_int(Y3),A4)
           => ( ( X != Y3 )
             => ( member5262025264175285858nt_int(product_Pair_int_int(X,Y3),R)
                | member5262025264175285858nt_int(product_Pair_int_int(Y3,X),R) ) ) ) )
     => total_on_int(A4,R) ) ).

% total_onI
tff(fact_6682_total__on__def,axiom,
    ! [A4: set_uint32,R: set_Pr1773385645901665561uint32] :
      ( total_on_uint32(A4,R)
    <=> ! [X5: uint32] :
          ( member_uint32(X5,A4)
         => ! [Xa3: uint32] :
              ( member_uint32(Xa3,A4)
             => ( ( X5 != Xa3 )
               => ( member8027108493173000802uint32(produc1400373151660368625uint32(X5,Xa3),R)
                  | member8027108493173000802uint32(produc1400373151660368625uint32(Xa3,X5),R) ) ) ) ) ) ).

% total_on_def
tff(fact_6683_total__on__def,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( total_on_nat(A4,R)
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => ! [Xa3: nat] :
              ( aa_set_nat_o(member_nat(Xa3),A4)
             => ( ( X5 != Xa3 )
               => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa3),R)
                  | member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa3),X5),R) ) ) ) ) ) ).

% total_on_def
tff(fact_6684_total__on__def,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int] :
      ( total_on_int(A4,R)
    <=> ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),A4)
             => ( ( X5 != Xa3 )
               => ( member5262025264175285858nt_int(product_Pair_int_int(X5,Xa3),R)
                  | member5262025264175285858nt_int(product_Pair_int_int(Xa3,X5),R) ) ) ) ) ) ).

% total_on_def
tff(fact_6685_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_int,S: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_pu(S2),S))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6686_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_complex,S: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => finite6551019134538273531omplex(collect_set_complex(aa_set8643480048548153067plex_o(aTP_Lamm_pv(S2),S))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6687_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_Code_integer,S: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aa_set4044230130517860621eger_o(aTP_Lamm_pw(S2),S))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6688_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_nat,S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_px(S2),S))) ) ).

% finite_if_eq_beyond_finite
tff(fact_6689_total__on__empty,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : total_on_nat(bot_bot_set_nat,R) ).

% total_on_empty
tff(fact_6690_total__on__empty,axiom,
    ! [R: set_Pr958786334691620121nt_int] : total_on_int(bot_bot_set_int,R) ).

% total_on_empty
tff(fact_6691_total__on__empty,axiom,
    ! [R: set_Product_prod_o_o] : total_on_o(bot_bot_set_o,R) ).

% total_on_empty
tff(fact_6692_Fpow__def,axiom,
    ! [A4: set_nat] : ( finite_Fpow_nat(A4) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_py,A4)) ) ).

% Fpow_def
tff(fact_6693_Fpow__def,axiom,
    ! [A4: set_complex] : ( finite_Fpow_complex(A4) = collect_set_complex(aTP_Lamm_pz(A4)) ) ).

% Fpow_def
tff(fact_6694_Fpow__def,axiom,
    ! [A4: set_Code_integer] : ( finite1532502677820914807nteger(A4) = collec574505750873337192nteger(aTP_Lamm_qa(A4)) ) ).

% Fpow_def
tff(fact_6695_Fpow__def,axiom,
    ! [A4: set_int] : ( finite_Fpow_int(A4) = collect_set_int(aTP_Lamm_qb(A4)) ) ).

% Fpow_def
tff(fact_6696_distinct__finite__set,axiom,
    ! [Xa: set_VEBT_VEBT] : finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aTP_Lamm_qc(Xa))) ).

% distinct_finite_set
tff(fact_6697_distinct__finite__set,axiom,
    ! [Xa: set_nat] : finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_qd(Xa))) ).

% distinct_finite_set
tff(fact_6698_distinct__finite__set,axiom,
    ! [Xa: set_real] : finite306553202115118035t_real(collect_list_real(aTP_Lamm_qe(Xa))) ).

% distinct_finite_set
tff(fact_6699_distinct__finite__set,axiom,
    ! [Xa: set_o] : finite_finite_list_o(collect_list_o(aTP_Lamm_qf(Xa))) ).

% distinct_finite_set
tff(fact_6700_diff__power__eq__sum,axiom,
    ! [Xa: complex,Na: nat,Y: complex] : ( aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xa),aa_nat_nat(suc,Na))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,Na))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xa),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_qg(Xa),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ) ).

% diff_power_eq_sum
tff(fact_6701_diff__power__eq__sum,axiom,
    ! [Xa: uint32,Na: nat,Y: uint32] : ( minus_minus_uint32(aa_nat_uint32(power_power_uint32(Xa),aa_nat_nat(suc,Na)),aa_nat_uint32(power_power_uint32(Y),aa_nat_nat(suc,Na))) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,minus_minus_uint32(Xa,Y)),groups833757482993574392uint32(aa_uin238090125092722531uint32(aa_nat7697544214703702448uint32(aTP_Lamm_qh(Xa),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ) ).

% diff_power_eq_sum
tff(fact_6702_diff__power__eq__sum,axiom,
    ! [Xa: int,Na: nat,Y: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xa),aa_nat_nat(suc,Na))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_qi(Xa),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ) ).

% diff_power_eq_sum
tff(fact_6703_diff__power__eq__sum,axiom,
    ! [Xa: real,Na: nat,Y: real] : ( aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(Xa),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_qj(Xa),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ) ).

% diff_power_eq_sum
tff(fact_6704_power__diff__sumr2,axiom,
    ! [Xa: complex,Na: nat,Y: complex] : ( aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xa),Na)),aa_nat_complex(power_power_complex(Y),Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xa),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_qk(Xa),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% power_diff_sumr2
tff(fact_6705_power__diff__sumr2,axiom,
    ! [Xa: uint32,Na: nat,Y: uint32] : ( minus_minus_uint32(aa_nat_uint32(power_power_uint32(Xa),Na),aa_nat_uint32(power_power_uint32(Y),Na)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,minus_minus_uint32(Xa,Y)),groups833757482993574392uint32(aa_uin238090125092722531uint32(aa_nat7697544214703702448uint32(aTP_Lamm_ql(Xa),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% power_diff_sumr2
tff(fact_6706_power__diff__sumr2,axiom,
    ! [Xa: int,Na: nat,Y: int] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xa),Na)),aa_nat_int(power_power_int(Y),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)),aa_set_nat_int(groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_qm(Xa),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% power_diff_sumr2
tff(fact_6707_power__diff__sumr2,axiom,
    ! [Xa: real,Na: nat,Y: real] : ( aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(Xa),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_qn(Xa),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% power_diff_sumr2
tff(fact_6708_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,B4: real,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,Sa)
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2)) )
       => ( ( groups2240296850493347238T_real(F,Sa) = B4 )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6709_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_real,F: fun_real_real,B4: real,I: real] :
      ( aa_set_real_o(finite_finite_real,Sa)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,Sa) = B4 )
         => ( aa_set_real_o(member_real(I),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6710_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_int,F: fun_int_real,B4: real,I: int] :
      ( aa_set_int_o(finite_finite_int,Sa)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,Sa) = B4 )
         => ( aa_set_int_o(member_int(I),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6711_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_complex,F: fun_complex_real,B4: real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sa)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,Sa) = B4 )
         => ( aa_set_complex_o(member_complex(I),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6712_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_Code_integer,F: fun_Co3960896333048504454r_real,B4: real,I: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Sa)
     => ( ! [I2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I2)) )
       => ( ( groups1270011288395367621r_real(F,Sa) = B4 )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6713_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,B4: rat,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,Sa)
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2)) )
       => ( ( groups136491112297645522BT_rat(F,Sa) = B4 )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6714_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_real,F: fun_real_rat,B4: rat,I: real] :
      ( aa_set_real_o(finite_finite_real,Sa)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,Sa) = B4 )
         => ( aa_set_real_o(member_real(I),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6715_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_nat,F: fun_nat_rat,B4: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,Sa)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,Sa) = B4 )
         => ( aa_set_nat_o(member_nat(I),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6716_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_int,F: fun_int_rat,B4: rat,I: int] :
      ( aa_set_int_o(finite_finite_int,Sa)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,Sa) = B4 )
         => ( aa_set_int_o(member_int(I),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6717_sum__nonneg__leq__bound,axiom,
    ! [Sa: set_complex,F: fun_complex_rat,B4: rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sa)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,Sa) = B4 )
         => ( aa_set_complex_o(member_complex(I),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6718_sum__nonneg__0,axiom,
    ! [Sa: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,Sa)
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,I2)) )
       => ( ( groups2240296850493347238T_real(F,Sa) = zero_zero_real )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),Sa)
           => ( aa_VEBT_VEBT_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6719_sum__nonneg__0,axiom,
    ! [Sa: set_real,F: fun_real_real,I: real] :
      ( aa_set_real_o(finite_finite_real,Sa)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,Sa) = zero_zero_real )
         => ( aa_set_real_o(member_real(I),Sa)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6720_sum__nonneg__0,axiom,
    ! [Sa: set_int,F: fun_int_real,I: int] :
      ( aa_set_int_o(finite_finite_int,Sa)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,Sa) = zero_zero_real )
         => ( aa_set_int_o(member_int(I),Sa)
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6721_sum__nonneg__0,axiom,
    ! [Sa: set_complex,F: fun_complex_real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sa)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,Sa) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(I),Sa)
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6722_sum__nonneg__0,axiom,
    ! [Sa: set_Code_integer,F: fun_Co3960896333048504454r_real,I: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Sa)
     => ( ! [I2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I2),Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(F,I2)) )
       => ( ( groups1270011288395367621r_real(F,Sa) = zero_zero_real )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I),Sa)
           => ( aa_Code_integer_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6723_sum__nonneg__0,axiom,
    ! [Sa: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,Sa)
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,I2)) )
       => ( ( groups136491112297645522BT_rat(F,Sa) = zero_zero_rat )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),Sa)
           => ( aa_VEBT_VEBT_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6724_sum__nonneg__0,axiom,
    ! [Sa: set_real,F: fun_real_rat,I: real] :
      ( aa_set_real_o(finite_finite_real,Sa)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,Sa) = zero_zero_rat )
         => ( aa_set_real_o(member_real(I),Sa)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6725_sum__nonneg__0,axiom,
    ! [Sa: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,Sa)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,Sa) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I),Sa)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6726_sum__nonneg__0,axiom,
    ! [Sa: set_int,F: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,Sa)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,Sa) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I),Sa)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6727_sum__nonneg__0,axiom,
    ! [Sa: set_complex,F: fun_complex_rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Sa)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),Sa)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,Sa) = zero_zero_rat )
         => ( aa_set_complex_o(member_complex(I),Sa)
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6728_add__mset__in__multiset,axiom,
    ! [M7: fun_list_nat_nat,A: list_nat] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_mm(M7)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_lis557685189327716851_nat_o(aTP_Lamm_qo(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_6729_add__mset__in__multiset,axiom,
    ! [M7: fun_set_nat_nat,A: set_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_mo(M7)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_qp(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_6730_add__mset__in__multiset,axiom,
    ! [M7: fun_nat_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_mq(M7)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qq(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_6731_add__mset__in__multiset,axiom,
    ! [M7: fun_int_nat,A: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_ms(M7)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_qr(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_6732_add__mset__in__multiset,axiom,
    ! [M7: fun_complex_nat,A: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_mu(M7)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_qs(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_6733_add__mset__in__multiset,axiom,
    ! [M7: fun_Code_integer_nat,A: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_mw(M7)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_qt(M7),A))) ) ).

% add_mset_in_multiset
tff(fact_6734_sum__fun__comp,axiom,
    ! [S2: set_VEBT_VEBT,R2: set_nat,G: fun_VEBT_VEBT_nat,F: fun_nat_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_nat_o(finite_finite_nat,R2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_VEBT_VEBT_nat(G,S2)),R2)
         => ( groups8325533452322294502uint32(aa_fun2276759601835508476uint32(aTP_Lamm_qu(G),F),S2) = groups833757482993574392uint32(aa_fun7322500112117231024uint32(aa_fun3572208589875229258uint32(aTP_Lamm_qv(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6735_sum__fun__comp,axiom,
    ! [S2: set_real,R2: set_nat,G: fun_real_nat,F: fun_nat_uint32] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_nat_o(finite_finite_nat,R2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_real_nat(G,S2)),R2)
         => ( groups5944083974425963860uint32(aa_fun5130233130708504716uint32(aTP_Lamm_qw(G),F),S2) = groups833757482993574392uint32(aa_fun7322500112117231024uint32(aa_fun8445541558549760422uint32(aTP_Lamm_qx(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6736_sum__fun__comp,axiom,
    ! [S2: set_literal,R2: set_nat,G: fun_literal_nat,F: fun_nat_uint32] :
      ( finite5847741373460823677iteral(S2)
     => ( aa_set_nat_o(finite_finite_nat,R2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_literal_nat(G,S2)),R2)
         => ( groups4542640294765070598uint32(aa_fun6956465913453778588uint32(aTP_Lamm_qy(G),F),S2) = groups833757482993574392uint32(aa_fun7322500112117231024uint32(aa_fun4475802048650532586uint32(aTP_Lamm_ra(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6737_sum__fun__comp,axiom,
    ! [S2: set_VEBT_VEBT,R2: set_complex,G: fun_VE5569070535936029429omplex,F: fun_complex_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,R2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(image_3793382806556112285omplex(G,S2)),R2)
         => ( groups8325533452322294502uint32(aa_fun4101646971871848350uint32(aTP_Lamm_rb(G),F),S2) = groups8736914816313324502uint32(aa_fun947202747272686700uint32(aa_fun7504237614304368612uint32(aTP_Lamm_rc(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6738_sum__fun__comp,axiom,
    ! [S2: set_real,R2: set_complex,G: fun_real_complex,F: fun_complex_uint32] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,R2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(image_real_complex(G,S2)),R2)
         => ( groups5944083974425963860uint32(aa_fun6757232495900413034uint32(aTP_Lamm_rd(G),F),S2) = groups8736914816313324502uint32(aa_fun947202747272686700uint32(aa_fun5489866774865184576uint32(aTP_Lamm_re(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6739_sum__fun__comp,axiom,
    ! [S2: set_literal,R2: set_complex,G: fun_literal_complex,F: fun_complex_uint32] :
      ( finite5847741373460823677iteral(S2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,R2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(image_5274195009022015549omplex(G,S2)),R2)
         => ( groups4542640294765070598uint32(aa_fun794436630273323070uint32(aTP_Lamm_rf(G),F),S2) = groups8736914816313324502uint32(aa_fun947202747272686700uint32(aa_fun7541052212593818756uint32(aTP_Lamm_rh(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6740_sum__fun__comp,axiom,
    ! [S2: set_VEBT_VEBT,R2: set_Code_integer,G: fun_VE9219535225822561060nteger,F: fun_Co1554742298434871494uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,R2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_2092689629700589388nteger(G,S2)),R2)
         => ( groups8325533452322294502uint32(aa_fun1617233732993516207uint32(aTP_Lamm_ri(G),F),S2) = groups8847630953604152069uint32(aa_fun8111848626357839434uint32(aa_fun1560612253975464305uint32(aTP_Lamm_rk(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6741_sum__fun__comp,axiom,
    ! [S2: set_real,R2: set_Code_integer,G: fun_re6978948838062957830nteger,F: fun_Co1554742298434871494uint32] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,R2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_4958697645175560720nteger(G,S2)),R2)
         => ( groups5944083974425963860uint32(aa_fun94539430827241881uint32(aTP_Lamm_rl(G),F),S2) = groups8847630953604152069uint32(aa_fun8111848626357839434uint32(aa_fun1633138777266163917uint32(aTP_Lamm_rn(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6742_sum__fun__comp,axiom,
    ! [S2: set_literal,R2: set_Code_integer,G: fun_li486744618827423812nteger,F: fun_Co1554742298434871494uint32] :
      ( finite5847741373460823677iteral(S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,R2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_4442872163159017964nteger(G,S2)),R2)
         => ( groups4542640294765070598uint32(aa_fun7307945177061598671uint32(aTP_Lamm_ro(G),F),S2) = groups8847630953604152069uint32(aa_fun8111848626357839434uint32(aa_fun5035814983738846737uint32(aTP_Lamm_rq(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6743_sum__fun__comp,axiom,
    ! [S2: set_nat,R2: set_nat,G: fun_nat_nat,F: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,R2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(image_nat_nat(G),S2)),R2)
         => ( groups833757482993574392uint32(aa_fun7322500112117231024uint32(aTP_Lamm_rr(G),F),S2) = groups833757482993574392uint32(aa_fun7322500112117231024uint32(aa_fun6985408296456604618uint32(aTP_Lamm_rt(S2),G),F),R2) ) ) ) ) ).

% sum_fun_comp
tff(fact_6744_diff__preserves__multiset,axiom,
    ! [M7: fun_list_nat_nat,N6: fun_list_nat_nat] :
      ( finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_mm(M7)))
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun7915899554530022_nat_o(aTP_Lamm_ru(M7),N6))) ) ).

% diff_preserves_multiset
tff(fact_6745_diff__preserves__multiset,axiom,
    ! [M7: fun_set_nat_nat,N6: fun_set_nat_nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aTP_Lamm_mo(M7)))
     => aa_set_set_nat_o(finite1152437895449049373et_nat,collect_set_nat(aa_fun1332161875654541722_nat_o(aTP_Lamm_rv(M7),N6))) ) ).

% diff_preserves_multiset
tff(fact_6746_diff__preserves__multiset,axiom,
    ! [M7: fun_nat_nat,N6: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_mq(M7)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_rw(M7),N6))) ) ).

% diff_preserves_multiset
tff(fact_6747_diff__preserves__multiset,axiom,
    ! [M7: fun_int_nat,N6: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_ms(M7)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_rx(M7),N6))) ) ).

% diff_preserves_multiset
tff(fact_6748_diff__preserves__multiset,axiom,
    ! [M7: fun_complex_nat,N6: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_mu(M7)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_ry(M7),N6))) ) ).

% diff_preserves_multiset
tff(fact_6749_diff__preserves__multiset,axiom,
    ! [M7: fun_Code_integer_nat,N6: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_mw(M7)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun4054834855839773292eger_o(aTP_Lamm_rz(M7),N6))) ) ).

% diff_preserves_multiset
tff(fact_6750_sum_Ogroup,axiom,
    ! [S2: set_nat,T2: set_complex,G: fun_nat_complex,H2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(image_nat_complex(G,S2)),T2)
         => ( groups5693394587270226106ex_nat(aa_fun8118263755821012566ex_nat(aa_fun5276372505856632910ex_nat(aTP_Lamm_sa(S2),G),H2),T2) = aa_set_nat_nat(groups3542108847815614940at_nat(H2),S2) ) ) ) ) ).

% sum.group
tff(fact_6751_sum_Ogroup,axiom,
    ! [S2: set_nat,T2: set_Code_integer,G: fun_nat_Code_integer,H2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_1215581382706833972nteger(G,S2)),T2)
         => ( groups7237345082560585321er_nat(aa_fun2720235278490593797er_nat(aa_fun676797813428960940er_nat(aTP_Lamm_sc(S2),G),H2),T2) = aa_set_nat_nat(groups3542108847815614940at_nat(H2),S2) ) ) ) ) ).

% sum.group
tff(fact_6752_sum_Ogroup,axiom,
    ! [S2: set_nat,T2: set_int,G: fun_nat_int,H2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_int_o(finite_finite_int,T2)
       => ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(G),S2)),T2)
         => ( groups4541462559716669496nt_nat(aa_fun6790803225114646484nt_nat(aa_fun5866268260113680714nt_nat(aTP_Lamm_se(S2),G),H2),T2) = aa_set_nat_nat(groups3542108847815614940at_nat(H2),S2) ) ) ) ) ).

% sum.group
tff(fact_6753_sum_Ogroup,axiom,
    ! [S2: set_complex,T2: set_nat,G: fun_complex_nat,H2: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,image_complex_nat(G,S2)),T2)
         => ( groups2073611262835488442omplex(aa_fun5031202185176517138omplex(aa_fun2450700647192873226omplex(aTP_Lamm_sf(S2),G),H2),T2) = groups7754918857620584856omplex(H2,S2) ) ) ) ) ).

% sum.group
tff(fact_6754_sum_Ogroup,axiom,
    ! [S2: set_complex,T2: set_Code_integer,G: fun_co8153928105989009288nteger,H2: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_1994230757181692690nteger(G,S2)),T2)
         => ( groups8024822376189712711omplex(aa_fun7233930908441553055omplex(aa_fun2168485499150143268omplex(aTP_Lamm_sh(S2),G),H2),T2) = groups7754918857620584856omplex(H2,S2) ) ) ) ) ).

% sum.group
tff(fact_6755_sum_Ogroup,axiom,
    ! [S2: set_complex,T2: set_int,G: fun_complex_int,H2: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( aa_set_int_o(finite_finite_int,T2)
       => ( aa_set_int_o(ord_less_eq_set_int(image_complex_int(G,S2)),T2)
         => ( groups3049146728041665814omplex(aa_fun6006737650382694510omplex(aa_fun5961093689379013058omplex(aTP_Lamm_sj(S2),G),H2),T2) = groups7754918857620584856omplex(H2,S2) ) ) ) ) ).

% sum.group
tff(fact_6756_sum_Ogroup,axiom,
    ! [S2: set_nat,T2: set_complex,G: fun_nat_complex,H2: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(image_nat_complex(G,S2)),T2)
         => ( groups5808333547571424918x_real(aa_fun2896631202927662350x_real(aa_fun3645564257361631750x_real(aTP_Lamm_sk(S2),G),H2),T2) = groups6591440286371151544t_real(H2,S2) ) ) ) ) ).

% sum.group
tff(fact_6757_sum_Ogroup,axiom,
    ! [S2: set_nat,T2: set_Code_integer,G: fun_nat_Code_integer,H2: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T2)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(image_1215581382706833972nteger(G,S2)),T2)
         => ( groups1270011288395367621r_real(aa_fun8082968229890050365r_real(aa_fun4476568600492114404r_real(aTP_Lamm_sl(S2),G),H2),T2) = groups6591440286371151544t_real(H2,S2) ) ) ) ) ).

% sum.group
tff(fact_6758_sum_Ogroup,axiom,
    ! [S2: set_nat,T2: set_int,G: fun_nat_int,H2: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_int_o(finite_finite_int,T2)
       => ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(G),S2)),T2)
         => ( groups8778361861064173332t_real(aa_fun4026127921582108556t_real(aa_fun4107705952538744322t_real(aTP_Lamm_sm(S2),G),H2),T2) = groups6591440286371151544t_real(H2,S2) ) ) ) ) ).

% sum.group
tff(fact_6759_sum_Ogroup,axiom,
    ! [S2: set_int,T2: set_nat,G: fun_int_nat,H2: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_int_set_nat(image_int_nat(G),S2)),T2)
         => ( aa_set_nat_int(groups3539618377306564664at_int(aa_fun5280992047749994636at_int(aa_fun4682031307293100034at_int(aTP_Lamm_sn(S2),G),H2)),T2) = groups4538972089207619220nt_int(H2,S2) ) ) ) ) ).

% sum.group
tff(fact_6760_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups5712668689793887828uint32(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_so(G)))) = groups5712668689793887828uint32(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6761_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups8736914816313324502uint32(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_sp(G)))) = groups8736914816313324502uint32(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6762_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups8847630953604152069uint32(G,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_sq(G)))) = groups8847630953604152069uint32(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6763_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_sr(G)))) = groups8778361861064173332t_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6764_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_ss(G)))) = groups5808333547571424918x_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6765_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_st(G)))) = groups1270011288395367621r_real(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6766_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_su(G)))) = groups3906332499630173760nt_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6767_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A4,collect_complex(aTP_Lamm_sv(G)))) = groups5058264527183730370ex_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6768_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A4,collect_Code_integer(aTP_Lamm_sw(G)))) = groups6602215022474089585er_rat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6769_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A4,collect_int(aTP_Lamm_sx(G)))) = groups4541462559716669496nt_nat(G,A4) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6770_sum__Suc,axiom,
    ! [F: fun_complex_nat,A4: set_complex] : ( groups5693394587270226106ex_nat(aTP_Lamm_sy(F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(F,A4)),finite_card_complex(A4)) ) ).

% sum_Suc
tff(fact_6771_sum__Suc,axiom,
    ! [F: fun_literal_nat,A4: set_literal] : ( groups8652099787943017962al_nat(aTP_Lamm_sz(F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8652099787943017962al_nat(F,A4)),finite_card_literal(A4)) ) ).

% sum_Suc
tff(fact_6772_sum__Suc,axiom,
    ! [F: fun_list_nat_nat,A4: set_list_nat] : ( groups4396056296759096172at_nat(aTP_Lamm_ta(F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4396056296759096172at_nat(F,A4)),finite_card_list_nat(A4)) ) ).

% sum_Suc
tff(fact_6773_sum__Suc,axiom,
    ! [F: fun_set_nat_nat,A4: set_set_nat] : ( groups8294997508430121362at_nat(aTP_Lamm_tb(F),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups8294997508430121362at_nat(F,A4)),finite_card_set_nat(A4)) ) ).

% sum_Suc
tff(fact_6774_sum__Suc,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_tc(F)),A4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A4)),aa_set_nat_nat(finite_card_nat,A4)) ) ).

% sum_Suc
tff(fact_6775_sum__multicount,axiom,
    ! [S2: set_VEBT_VEBT,T2: set_VEBT_VEBT,R2: fun_VE2962686792283380846VEBT_o,K: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T2)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),T2)
             => ( finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_VEB1521802747257091359VEBT_o(aa_fun8771395773580142140VEBT_o(aTP_Lamm_or(S2),R2),X))) = K ) )
         => ( groups771621172384141258BT_nat(aa_fun2100738023942440677BT_nat(aTP_Lamm_ot(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite7802652506058667612T_VEBT(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6776_sum__multicount,axiom,
    ! [S2: set_VEBT_VEBT,T2: set_real,R2: fun_VE3823611782367660756real_o,K: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_real_o(finite_finite_real,T2)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),T2)
             => ( finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_rea6225614514526338833VEBT_o(aa_fun575193042032787426VEBT_o(aTP_Lamm_ou(S2),R2),X))) = K ) )
         => ( groups771621172384141258BT_nat(aa_fun2935073076783396613BT_nat(aTP_Lamm_ow(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_real(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6777_sum__multicount,axiom,
    ! [S2: set_real,T2: set_VEBT_VEBT,R2: fun_re7829509182167823952VEBT_o,K: nat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T2)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),T2)
             => ( finite_card_real(collect_real(aa_VEB641519613517496269real_o(aa_fun7937150527584618978real_o(aTP_Lamm_ox(S2),R2),X))) = K ) )
         => ( groups1935376822645274424al_nat(aa_fun1148221025137894919al_nat(aTP_Lamm_oz(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite7802652506058667612T_VEBT(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6778_sum__multicount,axiom,
    ! [S2: set_real,T2: set_real,R2: fun_real_fun_real_o,K: nat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( aa_set_real_o(finite_finite_real,T2)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),T2)
             => ( finite_card_real(collect_real(aa_real_fun_real_o(aa_fun4467907209059791804real_o(aTP_Lamm_pa(S2),R2),X))) = K ) )
         => ( groups1935376822645274424al_nat(aa_fun3429493660699118127al_nat(aTP_Lamm_pc(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_real(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6779_sum__multicount,axiom,
    ! [S2: set_VEBT_VEBT,T2: set_literal,R2: fun_VE8371440389858774350eral_o,K: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( finite5847741373460823677iteral(T2)
       => ( ! [X: literal] :
              ( member_literal(X,T2)
             => ( finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_lit1284886718724587071VEBT_o(aa_fun5379558016612791804VEBT_o(aTP_Lamm_pd(S2),R2),X))) = K ) )
         => ( groups771621172384141258BT_nat(aa_fun3014642116068820613BT_nat(aTP_Lamm_pf(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_literal(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6780_sum__multicount,axiom,
    ! [S2: set_real,T2: set_literal,R2: fun_re3850508403959675952eral_o,K: nat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( finite5847741373460823677iteral(T2)
       => ( ! [X: literal] :
              ( member_literal(X,T2)
             => ( finite_card_real(collect_real(aa_lit1368651074454191789real_o(aa_fun3100033197786232354real_o(aTP_Lamm_pg(S2),R2),X))) = K ) )
         => ( groups1935376822645274424al_nat(aa_fun705237418523604071al_nat(aTP_Lamm_pi(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_literal(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6781_sum__multicount,axiom,
    ! [S2: set_literal,T2: set_VEBT_VEBT,R2: fun_li8107907266037299598VEBT_o,K: nat] :
      ( finite5847741373460823677iteral(S2)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T2)
       => ( ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),T2)
             => ( finite_card_literal(collect_literal(aa_VEB1719478621182999551eral_o(aa_fun1102782463918247292eral_o(aTP_Lamm_pj(S2),R2),X))) = K ) )
         => ( groups8652099787943017962al_nat(aa_fun1876825894470965861al_nat(aTP_Lamm_pl(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite7802652506058667612T_VEBT(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6782_sum__multicount,axiom,
    ! [S2: set_literal,T2: set_real,R2: fun_li5129792558484764084real_o,K: nat] :
      ( finite5847741373460823677iteral(S2)
     => ( aa_set_real_o(finite_finite_real,T2)
       => ( ! [X: real] :
              ( aa_set_real_o(member_real(X),T2)
             => ( finite_card_literal(collect_literal(aa_rea1838623199774720753eral_o(aa_fun5708556627802011170eral_o(aTP_Lamm_pm(S2),R2),X))) = K ) )
         => ( groups8652099787943017962al_nat(aa_fun202459941351508933al_nat(aTP_Lamm_po(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_real(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6783_sum__multicount,axiom,
    ! [S2: set_literal,T2: set_literal,R2: fun_li4177005666941401710eral_o,K: nat] :
      ( finite5847741373460823677iteral(S2)
     => ( finite5847741373460823677iteral(T2)
       => ( ! [X: literal] :
              ( member_literal(X,T2)
             => ( finite_card_literal(collect_literal(aa_lit5986169069484365087eral_o(aa_fun6827806182750346556eral_o(aTP_Lamm_pp(S2),R2),X))) = K ) )
         => ( groups8652099787943017962al_nat(aa_fun8110944661589842437al_nat(aTP_Lamm_pr(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),finite_card_literal(T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6784_sum__multicount,axiom,
    ! [S2: set_VEBT_VEBT,T2: set_nat,R2: fun_VE6480751511521678896_nat_o,K: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),T2)
             => ( finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_ft(S2),R2),X))) = K ) )
         => ( groups771621172384141258BT_nat(aa_fun3621025987637128233BT_nat(aTP_Lamm_td(T2),R2),S2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_set_nat_nat(finite_card_nat,T2)) ) ) ) ) ).

% sum_multicount
tff(fact_6785_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_te(M7),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_6786_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_tf(M7),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_te(M7),I))) ) ) ).

% card_less_Suc
tff(fact_6787_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_tf(M7),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_te(M7),I))) ) ) ).

% card_less_Suc2
tff(fact_6788_foldr__length__aux,axiom,
    ! [L: list_VEBT_VEBT,A: nat] : ( aa_nat_nat(foldr_VEBT_VEBT_nat(aTP_Lamm_dc,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_s6755466524823107622T_VEBT(L)) ) ).

% foldr_length_aux
tff(fact_6789_foldr__length__aux,axiom,
    ! [L: list_real,A: nat] : ( aa_nat_nat(foldr_real_nat(aTP_Lamm_dd,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_real(L)) ) ).

% foldr_length_aux
tff(fact_6790_foldr__length__aux,axiom,
    ! [L: list_o,A: nat] : ( aa_nat_nat(foldr_o_nat(aTP_Lamm_de,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),size_size_list_o(L)) ) ).

% foldr_length_aux
tff(fact_6791_foldr__length__aux,axiom,
    ! [L: list_nat,A: nat] : ( aa_nat_nat(foldr_nat_nat(aTP_Lamm_df,L),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_list_nat_nat(size_size_list_nat,L)) ) ).

% foldr_length_aux
tff(fact_6792_bot_Oordering__top__axioms,axiom,
    aa_set_nat_o(ordering_top_set_nat(aTP_Lamm_cr,aTP_Lamm_fb),bot_bot_set_nat) ).

% bot.ordering_top_axioms
tff(fact_6793_bot_Oordering__top__axioms,axiom,
    ordering_top_assn(aTP_Lamm_tg,aTP_Lamm_th,bot_bot_assn) ).

% bot.ordering_top_axioms
tff(fact_6794_bot_Oordering__top__axioms,axiom,
    aa_set_o_o(ordering_top_set_o(aTP_Lamm_ti,aTP_Lamm_tj),bot_bot_set_o) ).

% bot.ordering_top_axioms
tff(fact_6795_bot_Oordering__top__axioms,axiom,
    aa_set_int_o(ordering_top_set_int(aTP_Lamm_cu,aTP_Lamm_tk),bot_bot_set_int) ).

% bot.ordering_top_axioms
tff(fact_6796_bot_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_bs,aTP_Lamm_br),bot_bot_nat) ).

% bot.ordering_top_axioms
tff(fact_6797_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aq),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aq),S2)) ).

% card_sum_le_nat_sum
tff(fact_6798_set__zip__rightD,axiom,
    ! [Xa: $o,Y: $o,Xs: list_o,Ys: list_o] :
      ( member7466972457876170832od_o_o(product_Pair_o_o((Xa),(Y)),set_Product_prod_o_o2(zip_o_o(Xs,Ys)))
     => aa_set_o_o(member_o((Y)),aa_list_o_set_o(set_o2,Ys)) ) ).

% set_zip_rightD
tff(fact_6799_set__zip__rightD,axiom,
    ! [Xa: uint32,Y: uint32,Xs: list_uint32,Ys: list_uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y),set_Pr2418681094996576974uint32(zip_uint32_uint32(Xs,Ys)))
     => member_uint32(Y,set_uint322(Ys)) ) ).

% set_zip_rightD
tff(fact_6800_set__zip__rightD,axiom,
    ! [Xa: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys)))
     => aa_set_nat_o(member_nat(Y),aa_list_nat_set_nat(set_nat2,Ys)) ) ).

% set_zip_rightD
tff(fact_6801_set__zip__rightD,axiom,
    ! [Xa: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => aa_set_int_o(member_int(Y),aa_list_int_set_int(set_int2,Ys)) ) ).

% set_zip_rightD
tff(fact_6802_set__zip__rightD,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,Xs: list_f5222602229612168821_nat_o,Ys: list_P9062070895058802706et_nat] :
      ( member4513763875858039863et_nat(produc6017135503889078682et_nat(Xa,Y),set_Pr8131104793812927691et_nat(zip_fu1498620252286093946et_nat(Xs,Ys)))
     => member6260224972018164377et_nat(Y,set_Pr3864771053587467565et_nat(Ys)) ) ).

% set_zip_rightD
tff(fact_6803_set__zip__rightD,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,Xs: list_f5222602229612168821_nat_o,Ys: list_P8836981409019564976et_nat] :
      ( member9174161442226468181et_nat(produc7291855603632349880et_nat(Xa,Y),set_Pr6610751079997246441et_nat(zip_fu1756369540273021848et_nat(Xs,Ys)))
     => member4513763875858039863et_nat(Y,set_Pr8131104793812927691et_nat(Ys)) ) ).

% set_zip_rightD
tff(fact_6804_set__zip__leftD,axiom,
    ! [Xa: $o,Y: $o,Xs: list_o,Ys: list_o] :
      ( member7466972457876170832od_o_o(product_Pair_o_o((Xa),(Y)),set_Product_prod_o_o2(zip_o_o(Xs,Ys)))
     => aa_set_o_o(member_o((Xa)),aa_list_o_set_o(set_o2,Xs)) ) ).

% set_zip_leftD
tff(fact_6805_set__zip__leftD,axiom,
    ! [Xa: uint32,Y: uint32,Xs: list_uint32,Ys: list_uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y),set_Pr2418681094996576974uint32(zip_uint32_uint32(Xs,Ys)))
     => member_uint32(Xa,set_uint322(Xs)) ) ).

% set_zip_leftD
tff(fact_6806_set__zip__leftD,axiom,
    ! [Xa: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys)))
     => aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs)) ) ).

% set_zip_leftD
tff(fact_6807_set__zip__leftD,axiom,
    ! [Xa: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs)) ) ).

% set_zip_leftD
tff(fact_6808_set__zip__leftD,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Y: produc3658429121746597890et_nat,Xs: list_f5222602229612168821_nat_o,Ys: list_P9062070895058802706et_nat] :
      ( member4513763875858039863et_nat(produc6017135503889078682et_nat(Xa,Y),set_Pr8131104793812927691et_nat(zip_fu1498620252286093946et_nat(Xs,Ys)))
     => member7544390268018397336_nat_o(Xa,set_fu8796970219976790276_nat_o(Xs)) ) ).

% set_zip_leftD
tff(fact_6809_set__zip__leftD,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Y: produc8799533983703829792et_nat,Xs: list_f5222602229612168821_nat_o,Ys: list_P8836981409019564976et_nat] :
      ( member9174161442226468181et_nat(produc7291855603632349880et_nat(Xa,Y),set_Pr6610751079997246441et_nat(zip_fu1756369540273021848et_nat(Xs,Ys)))
     => member7544390268018397336_nat_o(Xa,set_fu8796970219976790276_nat_o(Xs)) ) ).

% set_zip_leftD
tff(fact_6810_in__set__zipE,axiom,
    ! [Xa: int,Y: vEBT_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( member2056185340421749780T_VEBT(produc3329399203697025711T_VEBT(Xa,Y),set_Pr8714266321650254504T_VEBT(zip_int_VEBT_VEBT(Xs,Ys)))
     => ~ ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
         => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6811_in__set__zipE,axiom,
    ! [Xa: int,Y: nat,Xs: list_int,Ys: list_nat] :
      ( member216504246829706758nt_nat(product_Pair_int_nat(Xa,Y),set_Pr6647972299459129970nt_nat(zip_int_nat(Xs,Ys)))
     => ~ ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
         => ~ aa_set_nat_o(member_nat(Y),aa_list_nat_set_nat(set_nat2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6812_in__set__zipE,axiom,
    ! [Xa: int,Y: real,Xs: list_int,Ys: list_real] :
      ( member2744130022092475746t_real(produc801115645435158769t_real(Xa,Y),set_Pr112895574167722958t_real(zip_int_real(Xs,Ys)))
     => ~ ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
         => ~ aa_set_real_o(member_real(Y),aa_lis3769085826154824284t_real(set_real2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6813_in__set__zipE,axiom,
    ! [Xa: int,Y: $o,Xs: list_int,Ys: list_o] :
      ( member4489920277610959864_int_o(product_Pair_int_o(Xa,(Y)),set_Pr8694291782656941196_int_o(zip_int_o(Xs,Ys)))
     => ~ ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
         => ~ aa_set_o_o(member_o((Y)),aa_list_o_set_o(set_o2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6814_in__set__zipE,axiom,
    ! [Xa: vEBT_VEBT,Y: int,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( member5419026705395827622BT_int(produc736041933913180425BT_int(Xa,Y),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys)))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
         => ~ aa_set_int_o(member_int(Y),aa_list_int_set_int(set_int2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6815_in__set__zipE,axiom,
    ! [Xa: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(Xa,Y),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys)))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
         => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6816_in__set__zipE,axiom,
    ! [Xa: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( member373505688050248522BT_nat(produc738532404422230701BT_nat(Xa,Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys)))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
         => ~ aa_set_nat_o(member_nat(Y),aa_list_nat_set_nat(set_nat2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6817_in__set__zipE,axiom,
    ! [Xa: vEBT_VEBT,Y: real,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( member8675245146396747942T_real(produc8117437818029410057T_real(Xa,Y),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys)))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
         => ~ aa_set_real_o(member_real(Y),aa_lis3769085826154824284t_real(set_real2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6818_in__set__zipE,axiom,
    ! [Xa: vEBT_VEBT,Y: $o,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( member3307348790968139188VEBT_o(produc8721562602347293563VEBT_o(Xa,(Y)),set_Pr7708085864119495200VEBT_o(zip_VEBT_VEBT_o(Xs,Ys)))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
         => ~ aa_set_o_o(member_o((Y)),aa_list_o_set_o(set_o2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6819_in__set__zipE,axiom,
    ! [Xa: nat,Y: int,Xs: list_nat,Ys: list_int] :
      ( member4262671552274231302at_int(product_Pair_nat_int(Xa,Y),set_Pr1470767568048878706at_int(zip_nat_int(Xs,Ys)))
     => ~ ( aa_set_nat_o(member_nat(Xa),aa_list_nat_set_nat(set_nat2,Xs))
         => ~ aa_set_int_o(member_int(Y),aa_list_int_set_int(set_int2,Ys)) ) ) ).

% in_set_zipE
tff(fact_6820_zip__same,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A,B),set_Pr9040384385603167362et_nat(zip_set_nat_set_nat(Xs,Xs)))
    <=> ( aa_set_set_nat_o(member_set_nat(A),set_set_nat2(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_6821_zip__same,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member568628332442017744T_VEBT(produc537772716801021591T_VEBT(A,B),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Xs)))
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_6822_zip__same,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( member7849222048561428706l_real(produc4511245868158468465l_real(A,B),set_Pr5999470521830281550l_real(zip_real_real(Xs,Xs)))
    <=> ( aa_set_real_o(member_real(A),aa_lis3769085826154824284t_real(set_real2,Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_6823_zip__same,axiom,
    ! [A: $o,B: $o,Xs: list_o] :
      ( member7466972457876170832od_o_o(product_Pair_o_o((A),(B)),set_Product_prod_o_o2(zip_o_o(Xs,Xs)))
    <=> ( aa_set_o_o(member_o((A)),aa_list_o_set_o(set_o2,Xs))
        & ( (A)
        <=> (B) ) ) ) ).

% zip_same
tff(fact_6824_zip__same,axiom,
    ! [A: uint32,B: uint32,Xs: list_uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(A,B),set_Pr2418681094996576974uint32(zip_uint32_uint32(Xs,Xs)))
    <=> ( member_uint32(A,set_uint322(Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_6825_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Xs)))
    <=> ( aa_set_nat_o(member_nat(A),aa_list_nat_set_nat(set_nat2,Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_6826_zip__same,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A,B),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Xs)))
    <=> ( aa_set_int_o(member_int(A),aa_list_int_set_int(set_int2,Xs))
        & ( A = B ) ) ) ).

% zip_same
tff(fact_6827_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_0_le_iff
tff(fact_6828_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_0_le_iff
tff(fact_6829_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_0_le_iff
tff(fact_6830_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_0_le_iff
tff(fact_6831_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_0_le_iff
tff(fact_6832_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_6833_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_6834_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_6835_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_6836_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_6837_zip__update,axiom,
    ! [Xs: list_o,I: nat,Xa: $o,Ys: list_o,Y: $o] : ( zip_o_o(list_update_o(Xs,I,(Xa)),list_update_o(Ys,I,(Y))) = list_u1537252308907898773od_o_o(zip_o_o(Xs,Ys),I,product_Pair_o_o((Xa),(Y))) ) ).

% zip_update
tff(fact_6838_zip__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xa: vEBT_VEBTi,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] : ( zip_VE793581609497812771_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xa),list_u6098035379799741383_VEBTi(Ys,I,Y)) = list_u30558089781959097_VEBTi(zip_VE793581609497812771_VEBTi(Xs,Ys),I,produc436343169921013763_VEBTi(Xa,Y)) ) ).

% zip_update
tff(fact_6839_zip__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xa: vEBT_VEBTi,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] : ( zip_VE7413257051550508102T_VEBT(list_u6098035379799741383_VEBTi(Xs,I,Xa),list_u1324408373059187874T_VEBT(Ys,I,Y)) = list_u9044509791230035014T_VEBT(zip_VE7413257051550508102T_VEBT(Xs,Ys),I,produc7053807326796202854T_VEBT(Xa,Y)) ) ).

% zip_update
tff(fact_6840_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xa: vEBT_VEBT,Ys: list_VEBT_VEBTi,Y: vEBT_VEBTi] : ( zip_VE6444338338598820466_VEBTi(list_u1324408373059187874T_VEBT(Xs,I,Xa),list_u6098035379799741383_VEBTi(Ys,I,Y)) = list_u636002643828906794_VEBTi(zip_VE6444338338598820466_VEBTi(Xs,Ys),I,produc6084888613844515218_VEBTi(Xa,Y)) ) ).

% zip_update
tff(fact_6841_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xa: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] : ( zip_VE537291747668921783T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xa),list_u1324408373059187874T_VEBT(Ys,I,Y)) = list_u6961636818849549845T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys),I,produc537772716801021591T_VEBT(Xa,Y)) ) ).

% zip_update
tff(fact_6842_zip__update,axiom,
    ! [Xs: list_uint32,I: nat,Xa: uint32,Ys: list_uint32,Y: uint32] : ( zip_uint32_uint32(list_update_uint32(Xs,I,Xa),list_update_uint32(Ys,I,Y)) = list_u2546161786664092711uint32(zip_uint32_uint32(Xs,Ys),I,produc1400373151660368625uint32(Xa,Y)) ) ).

% zip_update
tff(fact_6843_zip__update,axiom,
    ! [Xs: list_nat,I: nat,Xa: nat,Ys: list_nat,Y: nat] : ( zip_nat_nat(list_update_nat(Xs,I,Xa),list_update_nat(Ys,I,Y)) = list_u6180841689913720943at_nat(zip_nat_nat(Xs,Ys),I,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y)) ) ).

% zip_update
tff(fact_6844_zip__update,axiom,
    ! [Xs: list_int,I: nat,Xa: int,Ys: list_int,Y: int] : ( zip_int_int(list_update_int(Xs,I,Xa),list_update_int(Ys,I,Y)) = list_u3002344382305578791nt_int(zip_int_int(Xs,Ys),I,product_Pair_int_int(Xa,Y)) ) ).

% zip_update
tff(fact_6845_zip__update,axiom,
    ! [Xs: list_f5222602229612168821_nat_o,I: nat,Xa: fun_Pr6460944699146269935_nat_o,Ys: list_P9062070895058802706et_nat,Y: produc3658429121746597890et_nat] : ( zip_fu1498620252286093946et_nat(list_u2989257416974056029_nat_o(Xs,I,Xa),list_u9060326803697358356et_nat(Ys,I,Y)) = list_u4153759941229589042et_nat(zip_fu1498620252286093946et_nat(Xs,Ys),I,produc6017135503889078682et_nat(Xa,Y)) ) ).

% zip_update
tff(fact_6846_zip__update,axiom,
    ! [Xs: list_f5222602229612168821_nat_o,I: nat,Xa: fun_Pr6460944699146269935_nat_o,Ys: list_P8836981409019564976et_nat,Y: produc8799533983703829792et_nat] : ( zip_fu1756369540273021848et_nat(list_u2989257416974056029_nat_o(Xs,I,Xa),list_u4153759941229589042et_nat(Ys,I,Y)) = list_u3231058372064271824et_nat(zip_fu1756369540273021848et_nat(Xs,Ys),I,produc7291855603632349880et_nat(Xa,Y)) ) ).

% zip_update
tff(fact_6847_one__diff__power__eq_H,axiom,
    ! [Xa: rat,Na: nat] : ( aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xa),Na)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(one_one_rat),Xa)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_tl(Xa),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% one_diff_power_eq'
tff(fact_6848_one__diff__power__eq_H,axiom,
    ! [Xa: complex,Na: nat] : ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xa),Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xa)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_tm(Xa),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% one_diff_power_eq'
tff(fact_6849_one__diff__power__eq_H,axiom,
    ! [Xa: uint32,Na: nat] : ( minus_minus_uint32(one_one_uint32,aa_nat_uint32(power_power_uint32(Xa),Na)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,minus_minus_uint32(one_one_uint32,Xa)),groups833757482993574392uint32(aa_nat6935575117487981255uint32(aTP_Lamm_tn(Xa),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% one_diff_power_eq'
tff(fact_6850_one__diff__power__eq_H,axiom,
    ! [Xa: int,Na: nat] : ( aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xa),Na)) = 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),Xa)),aa_set_nat_int(groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_to(Xa),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% one_diff_power_eq'
tff(fact_6851_one__diff__power__eq_H,axiom,
    ! [Xa: real,Na: nat] : ( aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(one_one_real),Xa)),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_tp(Xa),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% one_diff_power_eq'
tff(fact_6852_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Na)) != zero_zero_rat ) ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_6853_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Na)) != zero_zero_nat ) ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_6854_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)) != zero_zero_real ) ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_6855_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) != zero_zero_int ) ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_6856_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Na: nat] : ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Na)) != zero_z3403309356797280102nteger ) ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_6857_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_6858_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_6859_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_6860_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_6861_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_6862_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_6863_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_6864_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ).

% less_imp_of_nat_less
tff(fact_6865_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% less_imp_of_nat_less
tff(fact_6866_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ) ).

% less_imp_of_nat_less
tff(fact_6867_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_6868_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I)),aa_nat_Code_integer(semiri4939895301339042750nteger,J)) ) ).

% of_nat_mono
tff(fact_6869_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_6870_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_6871_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_6872_Iio__eq__empty__iff,axiom,
    ! [Na: $o] :
      ( ( set_ord_lessThan_o((Na)) = bot_bot_set_o )
    <=> ( (Na)
      <=> bot_bot_o ) ) ).

% Iio_eq_empty_iff
tff(fact_6873_Iio__eq__empty__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Na) = bot_bot_set_nat )
    <=> ( Na = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_6874_lessThan__atLeast0,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_ord_lessThan_nat,Na) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na) ) ).

% lessThan_atLeast0
tff(fact_6875_lessThan__strict__subset__iff,axiom,
    ! [M: rat,Na: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_ord_lessThan_rat(M)),set_ord_lessThan_rat(Na))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_6876_lessThan__strict__subset__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_set_num_o(ord_less_set_num(set_ord_lessThan_num(M)),set_ord_lessThan_num(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_6877_lessThan__strict__subset__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_6878_lessThan__strict__subset__iff,axiom,
    ! [M: int,Na: int] :
      ( aa_set_int_o(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_6879_lessThan__strict__subset__iff,axiom,
    ! [M: real,Na: real] :
      ( aa_set_real_o(ord_less_set_real(set_or5984915006950818249n_real(M)),set_or5984915006950818249n_real(Na))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_6880_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Na) = bot_bot_set_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_6881_lessThan__Suc,axiom,
    ! [K: nat] : ( aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(insert_nat(K),aa_nat_set_nat(set_ord_lessThan_nat,K)) ) ).

% lessThan_Suc
tff(fact_6882_sum__Suc__diff_H,axiom,
    ! [M: nat,Na: nat,F: fun_nat_uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups833757482993574392uint32(aTP_Lamm_ks(F),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = minus_minus_uint32(aa_nat_uint32(F,Na),aa_nat_uint32(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_6883_sum__Suc__diff_H,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_int(groups3539618377306564664at_int(aTP_Lamm_kt(F)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Na)),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_6884_sum__Suc__diff_H,axiom,
    ! [M: nat,Na: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups6591440286371151544t_real(aTP_Lamm_ku(F),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) = aa_real_real(minus_minus_real(aa_nat_real(F,Na)),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff'
tff(fact_6885_Max__add__commute,axiom,
    ! [S2: set_complex,F: fun_complex_real,K: real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( lattic4275903605611617917x_real(image_complex_real(aa_rea4562580341872191937x_real(aTP_Lamm_tq(F),K),S2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic4275903605611617917x_real(image_complex_real(F,S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6886_Max__add__commute,axiom,
    ! [S2: set_Code_integer,F: fun_Co3960896333048504454r_real,K: real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ( lattic4275903605611617917x_real(image_7738145705984076560r_real(aa_rea8190550742337927152r_real(aTP_Lamm_tr(F),K),S2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic4275903605611617917x_real(image_7738145705984076560r_real(F,S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6887_Max__add__commute,axiom,
    ! [S2: set_complex,F: fun_complex_rat,K: rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( lattic7630753665789217321ax_rat(image_complex_rat(aa_rat15117248669498777ex_rat(aTP_Lamm_ts(F),K),S2)) = aa_rat_rat(plus_plus_rat(lattic7630753665789217321ax_rat(image_complex_rat(F,S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6888_Max__add__commute,axiom,
    ! [S2: set_Code_integer,F: fun_Code_integer_rat,K: rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ( lattic7630753665789217321ax_rat(image_315895873841295420er_rat(aa_rat6553461586499541832er_rat(aTP_Lamm_tt(F),K),S2)) = aa_rat_rat(plus_plus_rat(lattic7630753665789217321ax_rat(image_315895873841295420er_rat(F,S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6889_Max__add__commute,axiom,
    ! [S2: set_complex,F: fun_complex_int,K: int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( lattic8263393255366662781ax_int(image_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_tu(F),K),S2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,lattic8263393255366662781ax_int(image_complex_int(F,S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6890_Max__add__commute,axiom,
    ! [S2: set_Code_integer,F: fun_Code_integer_int,K: int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ( lattic8263393255366662781ax_int(image_948535463418740880er_int(aa_int1390634949194299888er_int(aTP_Lamm_tv(F),K),S2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,lattic8263393255366662781ax_int(image_948535463418740880er_int(F,S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6891_Max__add__commute,axiom,
    ! [S2: set_nat,F: fun_nat_real,K: real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( lattic4275903605611617917x_real(image_nat_real(aa_real_fun_nat_real(aTP_Lamm_tw(F),K),S2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic4275903605611617917x_real(image_nat_real(F,S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6892_Max__add__commute,axiom,
    ! [S2: set_nat,F: fun_nat_rat,K: rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( lattic7630753665789217321ax_rat(image_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_tx(F),K),S2)) = aa_rat_rat(plus_plus_rat(lattic7630753665789217321ax_rat(image_nat_rat(F,S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6893_Max__add__commute,axiom,
    ! [S2: set_nat,F: fun_nat_int,K: int] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( lattic8263393255366662781ax_int(aa_set_nat_set_int(image_nat_int(aa_int_fun_nat_int(aTP_Lamm_ty(F),K)),S2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,lattic8263393255366662781ax_int(aa_set_nat_set_int(image_nat_int(F),S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6894_Max__add__commute,axiom,
    ! [S2: set_int,F: fun_int_real,K: real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ( lattic4275903605611617917x_real(image_int_real(aa_real_fun_int_real(aTP_Lamm_tz(F),K),S2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic4275903605611617917x_real(image_int_real(F,S2))),K) ) ) ) ).

% Max_add_commute
tff(fact_6895_Min__add__commute,axiom,
    ! [S2: set_complex,F: fun_complex_real,K: real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( lattic3629708407755379051n_real(image_complex_real(aa_rea4562580341872191937x_real(aTP_Lamm_tq(F),K),S2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic3629708407755379051n_real(image_complex_real(F,S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6896_Min__add__commute,axiom,
    ! [S2: set_Code_integer,F: fun_Co3960896333048504454r_real,K: real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ( lattic3629708407755379051n_real(image_7738145705984076560r_real(aa_rea8190550742337927152r_real(aTP_Lamm_tr(F),K),S2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic3629708407755379051n_real(image_7738145705984076560r_real(F,S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6897_Min__add__commute,axiom,
    ! [S2: set_complex,F: fun_complex_rat,K: rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( lattic8086005427650270231in_rat(image_complex_rat(aa_rat15117248669498777ex_rat(aTP_Lamm_ts(F),K),S2)) = aa_rat_rat(plus_plus_rat(lattic8086005427650270231in_rat(image_complex_rat(F,S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6898_Min__add__commute,axiom,
    ! [S2: set_Code_integer,F: fun_Code_integer_rat,K: rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ( lattic8086005427650270231in_rat(image_315895873841295420er_rat(aa_rat6553461586499541832er_rat(aTP_Lamm_tt(F),K),S2)) = aa_rat_rat(plus_plus_rat(lattic8086005427650270231in_rat(image_315895873841295420er_rat(F,S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6899_Min__add__commute,axiom,
    ! [S2: set_complex,F: fun_complex_int,K: int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( lattic8718645017227715691in_int(image_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_tu(F),K),S2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,lattic8718645017227715691in_int(image_complex_int(F,S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6900_Min__add__commute,axiom,
    ! [S2: set_Code_integer,F: fun_Code_integer_int,K: int] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ( lattic8718645017227715691in_int(image_948535463418740880er_int(aa_int1390634949194299888er_int(aTP_Lamm_tv(F),K),S2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,lattic8718645017227715691in_int(image_948535463418740880er_int(F,S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6901_Min__add__commute,axiom,
    ! [S2: set_nat,F: fun_nat_real,K: real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( lattic3629708407755379051n_real(image_nat_real(aa_real_fun_nat_real(aTP_Lamm_tw(F),K),S2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic3629708407755379051n_real(image_nat_real(F,S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6902_Min__add__commute,axiom,
    ! [S2: set_nat,F: fun_nat_rat,K: rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( lattic8086005427650270231in_rat(image_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_tx(F),K),S2)) = aa_rat_rat(plus_plus_rat(lattic8086005427650270231in_rat(image_nat_rat(F,S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6903_Min__add__commute,axiom,
    ! [S2: set_nat,F: fun_nat_int,K: int] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( lattic8718645017227715691in_int(aa_set_nat_set_int(image_nat_int(aa_int_fun_nat_int(aTP_Lamm_ty(F),K)),S2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,lattic8718645017227715691in_int(aa_set_nat_set_int(image_nat_int(F),S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6904_Min__add__commute,axiom,
    ! [S2: set_int,F: fun_int_real,K: real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ( lattic3629708407755379051n_real(image_int_real(aa_real_fun_int_real(aTP_Lamm_tz(F),K),S2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,lattic3629708407755379051n_real(image_int_real(F,S2))),K) ) ) ) ).

% Min_add_commute
tff(fact_6905_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_nat,Na: nat,M: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or4665077453230672383an_nat(Na),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ua(G),Na),M)),aa_nat_set_nat(set_or4665077453230672383an_nat(Na),M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_6906_sum_OatLeastLessThan__rev,axiom,
    ! [G: fun_nat_real,Na: nat,M: nat] : ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or4665077453230672383an_nat(Na),M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_ub(G),Na),M),aa_nat_set_nat(set_or4665077453230672383an_nat(Na),M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_6907_of__nat__max,axiom,
    ! [Xa: nat,Y: nat] : ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xa)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ) ).

% of_nat_max
tff(fact_6908_of__nat__max,axiom,
    ! [Xa: nat,Y: nat] : ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y)) = aa_real_real(ord_max_real(aa_nat_real(semiri5074537144036343181t_real,Xa)),aa_nat_real(semiri5074537144036343181t_real,Y)) ) ).

% of_nat_max
tff(fact_6909_of__nat__max,axiom,
    ! [Xa: nat,Y: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y)) = aa_int_int(ord_max_int(aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(semiri1314217659103216013at_int,Y)) ) ).

% of_nat_max
tff(fact_6910_of__nat__max,axiom,
    ! [Xa: nat,Y: nat] : ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Y)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Xa)),aa_nat_Code_integer(semiri4939895301339042750nteger,Y)) ) ).

% of_nat_max
tff(fact_6911_finite__lists__length__eq,axiom,
    ! [A4: set_complex,Na: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_uc(A4),Na))) ) ).

% finite_lists_length_eq
tff(fact_6912_finite__lists__length__eq,axiom,
    ! [A4: set_Code_integer,Na: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_ud(A4),Na))) ) ).

% finite_lists_length_eq
tff(fact_6913_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,Na: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_ue(A4),Na))) ) ).

% finite_lists_length_eq
tff(fact_6914_finite__lists__length__eq,axiom,
    ! [A4: set_real,Na: nat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_uf(A4),Na))) ) ).

% finite_lists_length_eq
tff(fact_6915_finite__lists__length__eq,axiom,
    ! [A4: set_o,Na: nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_ug(A4),Na))) ) ).

% finite_lists_length_eq
tff(fact_6916_finite__lists__length__eq,axiom,
    ! [A4: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_uh(A4),Na))) ) ).

% finite_lists_length_eq
tff(fact_6917_finite__lists__length__eq,axiom,
    ! [A4: set_int,Na: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_ui(A4),Na))) ) ).

% finite_lists_length_eq
tff(fact_6918_distinct__finite__subset,axiom,
    ! [Xa: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,Xa)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aTP_Lamm_uj(Xa))) ) ).

% distinct_finite_subset
tff(fact_6919_distinct__finite__subset,axiom,
    ! [Xa: set_real] :
      ( aa_set_real_o(finite_finite_real,Xa)
     => finite306553202115118035t_real(collect_list_real(aTP_Lamm_uk(Xa))) ) ).

% distinct_finite_subset
tff(fact_6920_distinct__finite__subset,axiom,
    ! [Xa: set_o] :
      ( aa_set_o_o(finite_finite_o,Xa)
     => finite_finite_list_o(collect_list_o(aTP_Lamm_ul(Xa))) ) ).

% distinct_finite_subset
tff(fact_6921_distinct__finite__subset,axiom,
    ! [Xa: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,Xa)
     => finite8100373058378681591st_nat(collect_list_nat(aTP_Lamm_um(Xa))) ) ).

% distinct_finite_subset
tff(fact_6922_distinct__finite__subset,axiom,
    ! [Xa: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,Xa)
     => finite8712137658972009173omplex(collect_list_complex(aTP_Lamm_un(Xa))) ) ).

% distinct_finite_subset
tff(fact_6923_distinct__finite__subset,axiom,
    ! [Xa: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Xa)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aTP_Lamm_uo(Xa))) ) ).

% distinct_finite_subset
tff(fact_6924_distinct__finite__subset,axiom,
    ! [Xa: set_int] :
      ( aa_set_int_o(finite_finite_int,Xa)
     => finite3922522038869484883st_int(collect_list_int(aTP_Lamm_up(Xa))) ) ).

% distinct_finite_subset
tff(fact_6925_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [K2: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K2)) ) ).

% finite_nat_bounded
tff(fact_6926_finite__nat__iff__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_6927_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real,C2: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_uq(A),B),C2),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(B,A)),groups2240296850493347238T_real(C2,minus_5127226145743854075T_VEBT(S2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))),groups2240296850493347238T_real(C2,minus_5127226145743854075T_VEBT(S2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_6928_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_real,C2: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ur(A),B),C2),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(B,A)),groups8097168146408367636l_real(C2,minus_minus_set_real(S2,aa_set_real_set_real(insert_real(A),bot_bot_set_real)))),groups8097168146408367636l_real(C2,minus_minus_set_real(S2,aa_set_real_set_real(insert_real(A),bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_6929_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_real,C2: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_us(A),B),C2),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5808333547571424918x_real(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_6930_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Co3960896333048504454r_real,C2: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_ut(A),B),C2),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(B,A)),groups1270011288395367621r_real(C2,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups1270011288395367621r_real(C2,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_6931_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat,C2: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_uu(A),B),C2),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_rat_rat(plus_plus_rat(aa_VEBT_VEBT_rat(B,A)),groups136491112297645522BT_rat(C2,minus_5127226145743854075T_VEBT(S2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))),groups136491112297645522BT_rat(C2,minus_5127226145743854075T_VEBT(S2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_6932_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_rat,C2: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_uv(A),B),C2),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_rat_rat(plus_plus_rat(aa_real_rat(B,A)),groups1300246762558778688al_rat(C2,minus_minus_set_real(S2,aa_set_real_set_real(insert_real(A),bot_bot_set_real)))),groups1300246762558778688al_rat(C2,minus_minus_set_real(S2,aa_set_real_set_real(insert_real(A),bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_6933_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: fun_complex_rat,C2: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_uw(A),B),C2),S2) = $ite(aa_set_complex_o(member_complex(A),S2),aa_rat_rat(plus_plus_rat(aa_complex_rat(B,A)),groups5058264527183730370ex_rat(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))),groups5058264527183730370ex_rat(C2,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_6934_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: fun_Code_integer_rat,C2: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S2)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_ux(A),B),C2),S2) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A),S2),aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(B,A)),groups6602215022474089585er_rat(C2,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))),groups6602215022474089585er_rat(C2,minus_2355218937544613996nteger(S2,insert_Code_integer(A,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_6935_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_nat,C2: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S2)
     => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_uy(A),B),C2),S2) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(B,A)),groups771621172384141258BT_nat(C2,minus_5127226145743854075T_VEBT(S2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))),groups771621172384141258BT_nat(C2,minus_5127226145743854075T_VEBT(S2,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(A),bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_6936_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: fun_real_nat,C2: fun_real_nat] :
      ( aa_set_real_o(finite_finite_real,S2)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_uz(A),B),C2),S2) = $ite(aa_set_real_o(member_real(A),S2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(B,A)),groups1935376822645274424al_nat(C2,minus_minus_set_real(S2,aa_set_real_set_real(insert_real(A),bot_bot_set_real)))),groups1935376822645274424al_nat(C2,minus_minus_set_real(S2,aa_set_real_set_real(insert_real(A),bot_bot_set_real)))) ) ) ).

% sum.delta_remove
tff(fact_6937_card__lists__length__eq,axiom,
    ! [A4: set_literal,Na: nat] :
      ( finite5847741373460823677iteral(A4)
     => ( finite7554361855635516162iteral(collect_list_literal(aa_nat1159182957892526735eral_o(aTP_Lamm_va(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_literal(A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6938_card__lists__length__eq,axiom,
    ! [A4: set_list_nat,Na: nat] :
      ( finite8100373058378681591st_nat(A4)
     => ( finite7325466520557071688st_nat(collec5989764272469232197st_nat(aa_nat158606821896287091_nat_o(aTP_Lamm_vb(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_list_nat(A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6939_card__lists__length__eq,axiom,
    ! [A4: set_set_nat,Na: nat] :
      ( aa_set_set_nat_o(finite1152437895449049373et_nat,A4)
     => ( finite5631907774883551598et_nat(collect_list_set_nat(aa_nat2247512698059773005_nat_o(aTP_Lamm_vc(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_set_nat(A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6940_card__lists__length__eq,axiom,
    ! [A4: set_complex,Na: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => ( finite5120063068150530198omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_uc(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_complex(A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6941_card__lists__length__eq,axiom,
    ! [A4: set_Code_integer,Na: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => ( finite5823187341872139973nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_ud(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite4902975817058060853nteger(A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6942_card__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,Na: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => ( finite5915292604075114978T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_ue(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite7802652506058667612T_VEBT(A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6943_card__lists__length__eq,axiom,
    ! [A4: set_real,Na: nat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => ( finite4141703471298347796t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_uf(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_real(A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6944_card__lists__length__eq,axiom,
    ! [A4: set_o,Na: nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( finite_card_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_ug(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_o(A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6945_card__lists__length__eq,axiom,
    ! [A4: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_uh(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(finite_card_nat,A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6946_card__lists__length__eq,axiom,
    ! [A4: set_int,Na: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( finite_card_list_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_ui(A4),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,finite_card_int(A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_6947_finite__lists__length__le,axiom,
    ! [A4: set_complex,Na: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A4)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_vd(A4),Na))) ) ).

% finite_lists_length_le
tff(fact_6948_finite__lists__length__le,axiom,
    ! [A4: set_Code_integer,Na: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A4)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_ve(A4),Na))) ) ).

% finite_lists_length_le
tff(fact_6949_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,Na: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A4)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_vf(A4),Na))) ) ).

% finite_lists_length_le
tff(fact_6950_finite__lists__length__le,axiom,
    ! [A4: set_real,Na: nat] :
      ( aa_set_real_o(finite_finite_real,A4)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_vg(A4),Na))) ) ).

% finite_lists_length_le
tff(fact_6951_finite__lists__length__le,axiom,
    ! [A4: set_o,Na: nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_vh(A4),Na))) ) ).

% finite_lists_length_le
tff(fact_6952_finite__lists__length__le,axiom,
    ! [A4: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_vi(A4),Na))) ) ).

% finite_lists_length_le
tff(fact_6953_finite__lists__length__le,axiom,
    ! [A4: set_int,Na: nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_vj(A4),Na))) ) ).

% finite_lists_length_le
tff(fact_6954_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,Y: uint32] :
      ( ( size_s4844771616002835472uint32(Xs) = size_s4844771616002835472uint32(Ys) )
     => ( member_uint32(Y,set_uint322(Ys))
       => ~ ! [X: uint32] : ~ member8027108493173000802uint32(produc1400373151660368625uint32(X,Y),set_Pr2418681094996576974uint32(zip_uint32_uint32(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6955_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_int,Y: int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( aa_set_int_o(member_int(Y),aa_list_int_set_int(set_int2,Ys))
       => ~ ! [X: int] : ~ member5262025264175285858nt_int(product_Pair_int_int(X,Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6956_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,Y: int] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) )
     => ( aa_set_int_o(member_int(Y),aa_list_int_set_int(set_int2,Ys))
       => ~ ! [X: vEBT_VEBT] : ~ member5419026705395827622BT_int(produc736041933913180425BT_int(X,Y),set_Pr2853735649769556538BT_int(zip_VEBT_VEBT_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6957_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys))
       => ~ ! [X: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(X,Y),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6958_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,Y: real] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) )
     => ( aa_set_real_o(member_real(Y),aa_lis3769085826154824284t_real(set_real2,Ys))
       => ~ ! [X: vEBT_VEBT] : ~ member8675245146396747942T_real(produc8117437818029410057T_real(X,Y),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6959_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,Y: $o] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) )
     => ( aa_set_o_o(member_o((Y)),aa_list_o_set_o(set_o2,Ys))
       => ~ ! [X: vEBT_VEBT] : ~ member3307348790968139188VEBT_o(produc8721562602347293563VEBT_o(X,(Y)),set_Pr7708085864119495200VEBT_o(zip_VEBT_VEBT_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6960_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( aa_set_nat_o(member_nat(Y),aa_list_nat_set_nat(set_nat2,Ys))
       => ~ ! [X: vEBT_VEBT] : ~ member373505688050248522BT_nat(produc738532404422230701BT_nat(X,Y),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6961_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_int,Y: int] :
      ( ( size_size_list_real(Xs) = size_size_list_int(Ys) )
     => ( aa_set_int_o(member_int(Y),aa_list_int_set_int(set_int2,Ys))
       => ~ ! [X: real] : ~ member1627681773268152802al_int(produc3179012173361985393al_int(X,Y),set_Pr8219819362198175822al_int(zip_real_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6962_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( size_size_list_real(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Ys))
       => ~ ! [X: real] : ~ member7262085504369356948T_VEBT(produc6931449550656315951T_VEBT(X,Y),set_Pr8897343066327330088T_VEBT(zip_real_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6963_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_real,Ys: list_real,Y: real] :
      ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
     => ( aa_set_real_o(member_real(Y),aa_lis3769085826154824284t_real(set_real2,Ys))
       => ~ ! [X: real] : ~ member7849222048561428706l_real(produc4511245868158468465l_real(X,Y),set_Pr5999470521830281550l_real(zip_real_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6964_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,Xa: uint32] :
      ( ( size_s4844771616002835472uint32(Xs) = size_s4844771616002835472uint32(Ys) )
     => ( member_uint32(Xa,set_uint322(Xs))
       => ~ ! [Y3: uint32] : ~ member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y3),set_Pr2418681094996576974uint32(zip_uint32_uint32(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6965_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_int,Xa: int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
       => ~ ! [Y3: int] : ~ member5262025264175285858nt_int(product_Pair_int_int(Xa,Y3),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6966_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_VEBT_VEBT,Xa: int] :
      ( ( size_size_list_int(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
       => ~ ! [Y3: vEBT_VEBT] : ~ member2056185340421749780T_VEBT(produc3329399203697025711T_VEBT(Xa,Y3),set_Pr8714266321650254504T_VEBT(zip_int_VEBT_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6967_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_real,Xa: int] :
      ( ( size_size_list_int(Xs) = size_size_list_real(Ys) )
     => ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
       => ~ ! [Y3: real] : ~ member2744130022092475746t_real(produc801115645435158769t_real(Xa,Y3),set_Pr112895574167722958t_real(zip_int_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6968_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_o,Xa: int] :
      ( ( size_size_list_int(Xs) = size_size_list_o(Ys) )
     => ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
       => ~ ! [Y3: $o] : ~ member4489920277610959864_int_o(product_Pair_int_o(Xa,(Y3)),set_Pr8694291782656941196_int_o(zip_int_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6969_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_nat,Xa: int] :
      ( ( size_size_list_int(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( aa_set_int_o(member_int(Xa),aa_list_int_set_int(set_int2,Xs))
       => ~ ! [Y3: nat] : ~ member216504246829706758nt_nat(product_Pair_int_nat(Xa,Y3),set_Pr6647972299459129970nt_nat(zip_int_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6970_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Xa: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
       => ~ ! [Y3: vEBT_VEBT] : ~ member568628332442017744T_VEBT(produc537772716801021591T_VEBT(Xa,Y3),set_Pr9182192707038809660T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6971_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,Xa: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
       => ~ ! [Y3: real] : ~ member8675245146396747942T_real(produc8117437818029410057T_real(Xa,Y3),set_Pr1087130671499945274T_real(zip_VEBT_VEBT_real(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6972_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,Xa: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
       => ~ ! [Y3: $o] : ~ member3307348790968139188VEBT_o(produc8721562602347293563VEBT_o(Xa,(Y3)),set_Pr7708085864119495200VEBT_o(zip_VEBT_VEBT_o(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6973_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Xa: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_list_nat_nat(size_size_list_nat,Ys) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Xs))
       => ~ ! [Y3: nat] : ~ member373505688050248522BT_nat(produc738532404422230701BT_nat(Xa,Y3),set_Pr7031586669278753246BT_nat(zip_VEBT_VEBT_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6974_ex__less__of__nat__mult,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xa)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),Xa)) ) ).

% ex_less_of_nat_mult
tff(fact_6975_ex__less__of__nat__mult,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xa)) ) ).

% ex_less_of_nat_mult
tff(fact_6976_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_uint32(semiri2565882477558803405uint32,K) != zero_zero_uint32 )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_6977_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,K) != zero_zero_rat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_6978_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,K) != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_6979_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,K) != zero_zero_real )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_6980_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,K) != zero_zero_int )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_6981_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,K) != zero_z3403309356797280102nteger )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_6982_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_uint32(semiri2565882477558803405uint32,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = minus_minus_uint32(aa_nat_uint32(semiri2565882477558803405uint32,M),aa_nat_uint32(semiri2565882477558803405uint32,Na)) ) ) ).

% of_nat_diff
tff(fact_6983_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ) ).

% of_nat_diff
tff(fact_6984_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% of_nat_diff
tff(fact_6985_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ) ).

% of_nat_diff
tff(fact_6986_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ) ) ).

% of_nat_diff
tff(fact_6987_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
      ( ( zip_int_nat(Xs,Ys) = cons_P7512249878480867347nt_nat(Xy,Xys) )
     => ~ ! [X: int,Xs5: list_int] :
            ( ( Xs = cons_int(X,Xs5) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys = cons_nat(Y3,Ys4) )
               => ( ( Xy = product_Pair_int_nat(X,Y3) )
                 => ( Xys != zip_int_nat(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6988_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_o,Xy: product_prod_int_o,Xys: list_P5087981734274514673_int_o] :
      ( ( zip_int_o(Xs,Ys) = cons_P7321330006258091179_int_o(Xy,Xys) )
     => ~ ! [X: int,Xs5: list_int] :
            ( ( Xs = cons_int(X,Xs5) )
           => ! [Y3: $o,Ys4: list_o] :
                ( ( Ys = cons_o((Y3),Ys4) )
               => ( ( Xy = product_Pair_int_o(X,(Y3)) )
                 => ( Xys != zip_int_o(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6989_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
      ( ( zip_nat_int(Xs,Ys) = cons_P2335045147070616083at_int(Xy,Xys) )
     => ~ ! [X: nat,Xs5: list_nat] :
            ( ( Xs = cons_nat(X,Xs5) )
           => ! [Y3: int,Ys4: list_int] :
                ( ( Ys = cons_int(Y3,Ys4) )
               => ( ( Xy = product_Pair_nat_int(X,Y3) )
                 => ( Xys != zip_nat_int(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6990_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_o,Xy: product_prod_nat_o,Xys: list_P7333126701944960589_nat_o] :
      ( ( zip_nat_o(Xs,Ys) = cons_P9142372351690779143_nat_o(Xy,Xys) )
     => ~ ! [X: nat,Xs5: list_nat] :
            ( ( Xs = cons_nat(X,Xs5) )
           => ! [Y3: $o,Ys4: list_o] :
                ( ( Ys = cons_o((Y3),Ys4) )
               => ( ( Xy = product_Pair_nat_o(X,(Y3)) )
                 => ( Xys != zip_nat_o(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6991_zip__eq__ConsE,axiom,
    ! [Xs: list_o,Ys: list_int,Xy: product_prod_o_int,Xys: list_P3795440434834930179_o_int] :
      ( ( zip_o_int(Xs,Ys) = cons_P1455986808126089405_o_int(Xy,Xys) )
     => ~ ! [X: $o,Xs5: list_o] :
            ( ( Xs = cons_o((X),Xs5) )
           => ! [Y3: int,Ys4: list_int] :
                ( ( Ys = cons_int(Y3,Ys4) )
               => ( ( Xy = product_Pair_o_int((X),Y3) )
                 => ( Xys != zip_o_int(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6992_zip__eq__ConsE,axiom,
    ! [Xs: list_o,Ys: list_nat,Xy: product_prod_o_nat,Xys: list_P6285523579766656935_o_nat] :
      ( ( zip_o_nat(Xs,Ys) = cons_P5633837827635286113_o_nat(Xy,Xys) )
     => ~ ! [X: $o,Xs5: list_o] :
            ( ( Xs = cons_o((X),Xs5) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys = cons_nat(Y3,Ys4) )
               => ( ( Xy = product_Pair_o_nat((X),Y3) )
                 => ( Xys != zip_o_nat(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6993_zip__eq__ConsE,axiom,
    ! [Xs: list_o,Ys: list_o,Xy: product_prod_o_o,Xys: list_P4002435161011370285od_o_o] :
      ( ( zip_o_o(Xs,Ys) = cons_P8766293264717362397od_o_o(Xy,Xys) )
     => ~ ! [X: $o,Xs5: list_o] :
            ( ( Xs = cons_o((X),Xs5) )
           => ! [Y3: $o,Ys4: list_o] :
                ( ( Ys = cons_o((Y3),Ys4) )
               => ( ( Xy = product_Pair_o_o((X),(Y3)) )
                 => ( Xys != zip_o_o(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6994_zip__eq__ConsE,axiom,
    ! [Xs: list_uint32,Ys: list_uint32,Xy: produc827990862158126777uint32,Xys: list_P3069071885182933823uint32] :
      ( ( zip_uint32_uint32(Xs,Ys) = cons_P3149448846263281007uint32(Xy,Xys) )
     => ~ ! [X: uint32,Xs5: list_uint32] :
            ( ( Xs = cons_uint32(X,Xs5) )
           => ! [Y3: uint32,Ys4: list_uint32] :
                ( ( Ys = cons_uint32(Y3,Ys4) )
               => ( ( Xy = produc1400373151660368625uint32(X,Y3) )
                 => ( Xys != zip_uint32_uint32(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6995_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( zip_nat_nat(Xs,Ys) = cons_P6512896166579812791at_nat(Xy,Xys) )
     => ~ ! [X: nat,Xs5: list_nat] :
            ( ( Xs = cons_nat(X,Xs5) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys = cons_nat(Y3,Ys4) )
               => ( ( Xy = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3) )
                 => ( Xys != zip_nat_nat(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6996_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( zip_int_int(Xs,Ys) = cons_P3334398858971670639nt_int(Xy,Xys) )
     => ~ ! [X: int,Xs5: list_int] :
            ( ( Xs = cons_int(X,Xs5) )
           => ! [Y3: int,Ys4: list_int] :
                ( ( Ys = cons_int(Y3,Ys4) )
               => ( ( Xy = product_Pair_int_int(X,Y3) )
                 => ( Xys != zip_int_int(Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6997_lenlex__irreflexive,axiom,
    ! [R: set_Pr1773385645901665561uint32,Xs: list_uint32] :
      ( ! [X: uint32] : ~ member8027108493173000802uint32(produc1400373151660368625uint32(X,X),R)
     => ~ member2333554998283850498uint32(produc7487160679990061969uint32(Xs,Xs),lenlex_uint32(R)) ) ).

% lenlex_irreflexive
tff(fact_6998_lenlex__irreflexive,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X: nat] : ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X),R)
     => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Xs),lenlex_nat(R)) ) ).

% lenlex_irreflexive
tff(fact_6999_lenlex__irreflexive,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ! [X: int] : ~ member5262025264175285858nt_int(product_Pair_int_int(X,X),R)
     => ~ member6698963635872716290st_int(produc364263696895485585st_int(Xs,Xs),lenlex_int(R)) ) ).

% lenlex_irreflexive
tff(fact_7000_reals__Archimedean3,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ! [Y4: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xa)) ) ).

% reals_Archimedean3
tff(fact_7001_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xa: nat,Y: nat] :
      ( aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_vk(C2)),aa_nat_set_nat(set_or4665077453230672383an_nat(Xa),Y)) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),Y),
          aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C2)),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat),bot_bot_set_nat) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_7002_listrel1I1,axiom,
    ! [Xa: $o,Y: $o,R: set_Product_prod_o_o,Xs: list_o] :
      ( member7466972457876170832od_o_o(product_Pair_o_o((Xa),(Y)),R)
     => member4159035015898711888list_o(produc8435520187683070743list_o(cons_o((Xa),Xs),cons_o((Y),Xs)),listrel1_o(R)) ) ).

% listrel1I1
tff(fact_7003_listrel1I1,axiom,
    ! [Xa: uint32,Y: uint32,R: set_Pr1773385645901665561uint32,Xs: list_uint32] :
      ( member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y),R)
     => member2333554998283850498uint32(produc7487160679990061969uint32(cons_uint32(Xa,Xs),cons_uint32(Y,Xs)),listrel1_uint32(R)) ) ).

% listrel1I1
tff(fact_7004_listrel1I1,axiom,
    ! [Xa: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y),R)
     => member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(Xa,Xs),cons_nat(Y,Xs)),listrel1_nat(R)) ) ).

% listrel1I1
tff(fact_7005_listrel1I1,axiom,
    ! [Xa: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( member5262025264175285858nt_int(product_Pair_int_int(Xa,Y),R)
     => member6698963635872716290st_int(produc364263696895485585st_int(cons_int(Xa,Xs),cons_int(Y,Xs)),listrel1_int(R)) ) ).

% listrel1I1
tff(fact_7006_Cons__listrel1E1,axiom,
    ! [Xa: $o,Xs: list_o,Ys: list_o,R: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(cons_o((Xa),Xs),Ys),listrel1_o(R))
     => ( ! [Y3: $o] :
            ( ( Ys = cons_o((Y3),Xs) )
           => ~ member7466972457876170832od_o_o(product_Pair_o_o((Xa),(Y3)),R) )
       => ~ ! [Zs2: list_o] :
              ( ( Ys = cons_o((Xa),Zs2) )
             => ~ member4159035015898711888list_o(produc8435520187683070743list_o(Xs,Zs2),listrel1_o(R)) ) ) ) ).

% Cons_listrel1E1
tff(fact_7007_Cons__listrel1E1,axiom,
    ! [Xa: uint32,Xs: list_uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( member2333554998283850498uint32(produc7487160679990061969uint32(cons_uint32(Xa,Xs),Ys),listrel1_uint32(R))
     => ( ! [Y3: uint32] :
            ( ( Ys = cons_uint32(Y3,Xs) )
           => ~ member8027108493173000802uint32(produc1400373151660368625uint32(Xa,Y3),R) )
       => ~ ! [Zs2: list_uint32] :
              ( ( Ys = cons_uint32(Xa,Zs2) )
             => ~ member2333554998283850498uint32(produc7487160679990061969uint32(Xs,Zs2),listrel1_uint32(R)) ) ) ) ).

% Cons_listrel1E1
tff(fact_7008_Cons__listrel1E1,axiom,
    ! [Xa: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(Xa,Xs),Ys),listrel1_nat(R))
     => ( ! [Y3: nat] :
            ( ( Ys = cons_nat(Y3,Xs) )
           => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Y3),R) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys = cons_nat(Xa,Zs2) )
             => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Zs2),listrel1_nat(R)) ) ) ) ).

% Cons_listrel1E1
tff(fact_7009_Cons__listrel1E1,axiom,
    ! [Xa: int,Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(Xa,Xs),Ys),listrel1_int(R))
     => ( ! [Y3: int] :
            ( ( Ys = cons_int(Y3,Xs) )
           => ~ member5262025264175285858nt_int(product_Pair_int_int(Xa,Y3),R) )
       => ~ ! [Zs2: list_int] :
              ( ( Ys = cons_int(Xa,Zs2) )
             => ~ member6698963635872716290st_int(produc364263696895485585st_int(Xs,Zs2),listrel1_int(R)) ) ) ) ).

% Cons_listrel1E1
tff(fact_7010_Cons__listrel1E2,axiom,
    ! [Xs: list_o,Y: $o,Ys: list_o,R: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(Xs,cons_o((Y),Ys)),listrel1_o(R))
     => ( ! [X: $o] :
            ( ( Xs = cons_o((X),Ys) )
           => ~ member7466972457876170832od_o_o(product_Pair_o_o((X),(Y)),R) )
       => ~ ! [Zs2: list_o] :
              ( ( Xs = cons_o((Y),Zs2) )
             => ~ member4159035015898711888list_o(produc8435520187683070743list_o(Zs2,Ys),listrel1_o(R)) ) ) ) ).

% Cons_listrel1E2
tff(fact_7011_Cons__listrel1E2,axiom,
    ! [Xs: list_uint32,Y: uint32,Ys: list_uint32,R: set_Pr1773385645901665561uint32] :
      ( member2333554998283850498uint32(produc7487160679990061969uint32(Xs,cons_uint32(Y,Ys)),listrel1_uint32(R))
     => ( ! [X: uint32] :
            ( ( Xs = cons_uint32(X,Ys) )
           => ~ member8027108493173000802uint32(produc1400373151660368625uint32(X,Y),R) )
       => ~ ! [Zs2: list_uint32] :
              ( ( Xs = cons_uint32(Y,Zs2) )
             => ~ member2333554998283850498uint32(produc7487160679990061969uint32(Zs2,Ys),listrel1_uint32(R)) ) ) ) ).

% Cons_listrel1E2
tff(fact_7012_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,cons_nat(Y,Ys)),listrel1_nat(R))
     => ( ! [X: nat] :
            ( ( Xs = cons_nat(X,Ys) )
           => ~ member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),R) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs = cons_nat(Y,Zs2) )
             => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Zs2,Ys),listrel1_nat(R)) ) ) ) ).

% Cons_listrel1E2
tff(fact_7013_Cons__listrel1E2,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,cons_int(Y,Ys)),listrel1_int(R))
     => ( ! [X: int] :
            ( ( Xs = cons_int(X,Ys) )
           => ~ member5262025264175285858nt_int(product_Pair_int_int(X,Y),R) )
       => ~ ! [Zs2: list_int] :
              ( ( Xs = cons_int(Y,Zs2) )
             => ~ member6698963635872716290st_int(produc364263696895485585st_int(Zs2,Ys),listrel1_int(R)) ) ) ) ).

% Cons_listrel1E2
tff(fact_7014_sum__bounded__below,axiom,
    ! [A4: set_VEBT_VEBT,K4: real,F: fun_VEBT_VEBT_real] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K4),aa_VEBT_VEBT_real(F,I2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite7802652506058667612T_VEBT(A4))),K4)),groups2240296850493347238T_real(F,A4)) ) ).

% sum_bounded_below
tff(fact_7015_sum__bounded__below,axiom,
    ! [A4: set_real,K4: real,F: fun_real_real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K4),aa_real_real(F,I2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A4))),K4)),groups8097168146408367636l_real(F,A4)) ) ).

% sum_bounded_below
tff(fact_7016_sum__bounded__below,axiom,
    ! [A4: set_int,K4: real,F: fun_int_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K4),aa_int_real(F,I2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A4))),K4)),groups8778361861064173332t_real(F,A4)) ) ).

% sum_bounded_below
tff(fact_7017_sum__bounded__below,axiom,
    ! [A4: set_complex,K4: real,F: fun_complex_real] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K4),aa_complex_real(F,I2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A4))),K4)),groups5808333547571424918x_real(F,A4)) ) ).

% sum_bounded_below
tff(fact_7018_sum__bounded__below,axiom,
    ! [A4: set_literal,K4: real,F: fun_literal_real] :
      ( ! [I2: literal] :
          ( member_literal(I2,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K4),aa_literal_real(F,I2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_literal(A4))),K4)),groups2795228986978005958l_real(F,A4)) ) ).

% sum_bounded_below
tff(fact_7019_sum__bounded__below,axiom,
    ! [A4: set_VEBT_VEBT,K4: code_integer,F: fun_VE9219535225822561060nteger] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K4),aa_VEB5935388676523804373nteger(F,I2)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite7802652506058667612T_VEBT(A4))),K4)),groups5748017345553531991nteger(F,A4)) ) ).

% sum_bounded_below
tff(fact_7020_sum__bounded__below,axiom,
    ! [A4: set_real,K4: code_integer,F: fun_re6978948838062957830nteger] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K4),aa_real_Code_integer(F,I2)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_real(A4))),K4)),groups7713935264441627589nteger(F,A4)) ) ).

% sum_bounded_below
tff(fact_7021_sum__bounded__below,axiom,
    ! [A4: set_int,K4: code_integer,F: fun_int_Code_integer] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K4),aa_int_Code_integer(F,I2)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A4))),K4)),groups7873554091576472773nteger(F,A4)) ) ).

% sum_bounded_below
tff(fact_7022_sum__bounded__below,axiom,
    ! [A4: set_complex,K4: code_integer,F: fun_co8153928105989009288nteger] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K4),aa_com4217951221296830281nteger(F,I2)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_complex(A4))),K4)),groups6621422865394947399nteger(F,A4)) ) ).

% sum_bounded_below
tff(fact_7023_sum__bounded__below,axiom,
    ! [A4: set_literal,K4: code_integer,F: fun_li486744618827423812nteger] :
      ( ! [I2: literal] :
          ( member_literal(I2,A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K4),aa_lit3353659836003880949nteger(F,I2)) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_literal(A4))),K4)),groups8556826181031883383nteger(F,A4)) ) ).

% sum_bounded_below
tff(fact_7024_sum__bounded__above,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_real,K4: real] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F,I2)),K4) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2240296850493347238T_real(F,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite7802652506058667612T_VEBT(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7025_sum__bounded__above,axiom,
    ! [A4: set_real,F: fun_real_real,K4: real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I2)),K4) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_real(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7026_sum__bounded__above,axiom,
    ! [A4: set_int,F: fun_int_real,K4: real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I2)),K4) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_int(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7027_sum__bounded__above,axiom,
    ! [A4: set_complex,F: fun_complex_real,K4: real] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I2)),K4) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_complex(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7028_sum__bounded__above,axiom,
    ! [A4: set_literal,F: fun_literal_real,K4: real] :
      ( ! [I2: literal] :
          ( member_literal(I2,A4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_literal_real(F,I2)),K4) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2795228986978005958l_real(F,A4)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,finite_card_literal(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7029_sum__bounded__above,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger,K4: code_integer] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_VEB5935388676523804373nteger(F,I2)),K4) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups5748017345553531991nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite7802652506058667612T_VEBT(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7030_sum__bounded__above,axiom,
    ! [A4: set_real,F: fun_re6978948838062957830nteger,K4: code_integer] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_real_Code_integer(F,I2)),K4) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7713935264441627589nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_real(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7031_sum__bounded__above,axiom,
    ! [A4: set_int,F: fun_int_Code_integer,K4: code_integer] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(F,I2)),K4) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7873554091576472773nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7032_sum__bounded__above,axiom,
    ! [A4: set_complex,F: fun_co8153928105989009288nteger,K4: code_integer] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_com4217951221296830281nteger(F,I2)),K4) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups6621422865394947399nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_complex(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7033_sum__bounded__above,axiom,
    ! [A4: set_literal,F: fun_li486744618827423812nteger,K4: code_integer] :
      ( ! [I2: literal] :
          ( member_literal(I2,A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_lit3353659836003880949nteger(F,I2)),K4) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups8556826181031883383nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_literal(A4))),K4)) ) ).

% sum_bounded_above
tff(fact_7034_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M)) ) ).

% nat_less_real_le
tff(fact_7035_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_7036_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% lessThan_Suc_eq_insert_0
tff(fact_7037_lenlex__length,axiom,
    ! [Ms: list_VEBT_VEBT,Ns: list_VEBT_VEBT,R: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Ms,Ns),lenlex_VEBT_VEBT(R))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Ms)),size_s6755466524823107622T_VEBT(Ns)) ) ).

% lenlex_length
tff(fact_7038_lenlex__length,axiom,
    ! [Ms: list_real,Ns: list_real,R: set_Pr6218003697084177305l_real] :
      ( member6584958104391596930t_real(produc1408950526243324945t_real(Ms,Ns),lenlex_real(R))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Ms)),size_size_list_real(Ns)) ) ).

% lenlex_length
tff(fact_7039_lenlex__length,axiom,
    ! [Ms: list_o,Ns: list_o,R: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(Ms,Ns),lenlex_o(R))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Ms)),size_size_list_o(Ns)) ) ).

% lenlex_length
tff(fact_7040_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Ms,Ns),lenlex_nat(R))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Ms)),aa_list_nat_nat(size_size_list_nat,Ns)) ) ).

% lenlex_length
tff(fact_7041_real__archimedian__rdiv__eq__0,axiom,
    ! [Xa: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M3)),Xa)),C2) )
         => ( Xa = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_7042_sum__bounded__above__strict,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VEBT_VEBT_int,K4: int] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_VEBT_VEBT_int(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite7802652506058667612T_VEBT(A4))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups769130701875090982BT_int(F,A4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite7802652506058667612T_VEBT(A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7043_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F: fun_real_int,K4: int] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_real(A4))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups1932886352136224148al_int(F,A4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_real(A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7044_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F: fun_complex_int,K4: int] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_complex(A4))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_complex(A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7045_sum__bounded__above__strict,axiom,
    ! [A4: set_literal,F: fun_literal_int,K4: int] :
      ( ! [I2: literal] :
          ( member_literal(I2,A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_literal_int(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_literal(A4))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups8649609317433967686al_int(F,A4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,finite_card_literal(A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7046_sum__bounded__above__strict,axiom,
    ! [A4: set_nat,F: fun_nat_int,K4: int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(finite_card_nat,A4))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_set_nat_int(groups3539618377306564664at_int(F),A4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(finite_card_nat,A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7047_sum__bounded__above__strict,axiom,
    ! [A4: set_VEBT_VEBT,F: fun_VE9219535225822561060nteger,K4: code_integer] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_VEB5935388676523804373nteger(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite7802652506058667612T_VEBT(A4))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups5748017345553531991nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite7802652506058667612T_VEBT(A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7048_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F: fun_re6978948838062957830nteger,K4: code_integer] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_real_Code_integer(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_real(A4))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups7713935264441627589nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_real(A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7049_sum__bounded__above__strict,axiom,
    ! [A4: set_int,F: fun_int_Code_integer,K4: code_integer] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_int(A4))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups7873554091576472773nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_int(A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7050_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F: fun_co8153928105989009288nteger,K4: code_integer] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_com4217951221296830281nteger(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_complex(A4))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups6621422865394947399nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_complex(A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7051_sum__bounded__above__strict,axiom,
    ! [A4: set_literal,F: fun_li486744618827423812nteger,K4: code_integer] :
      ( ! [I2: literal] :
          ( member_literal(I2,A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_lit3353659836003880949nteger(F,I2)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),finite_card_literal(A4))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,groups8556826181031883383nteger(F,A4)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,finite_card_literal(A4))),K4)) ) ) ).

% sum_bounded_above_strict
tff(fact_7052_linear__plus__1__le__power,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),one_one_real)),Na)) ) ).

% linear_plus_1_le_power
tff(fact_7053_cnt__cnt__eq,axiom,
    ! [T: vEBT_VEBT] : ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,T)) ) ).

% cnt_cnt_eq
tff(fact_7054_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_minNulli(Tia),aa_VEB6687300313434771872o_assn(aTP_Lamm_vl(T),Tia)) ).

% minNulli_rule
tff(fact_7055_builupi_Hcorr,axiom,
    ! [Na: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))) ).

% builupi'corr
tff(fact_7056_builupicorr,axiom,
    ! [Na: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_vebt_buildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))) ).

% builupicorr
tff(fact_7057_vebt__maxtilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts2))
     => hoare_7629718768684598413on_nat(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts2,Tsi),vEBT_vebt_maxti(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Tsi),I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_vm(I),Ts2),Tsi)) ) ).

% vebt_maxtilist
tff(fact_7058_vebt__mintilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts2))
     => hoare_7629718768684598413on_nat(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts2,Tsi),vEBT_vebt_minti(aa_nat_VEBT_VEBTi(nth_VEBT_VEBTi(Tsi),I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_vn(I),Ts2),Tsi)) ) ).

% vebt_mintilist
tff(fact_7059_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_7060_negative__eq__positive,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_7061_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_minti(Tia),aa_VEB7611521244314475576t_assn(aTP_Lamm_vo(T),Tia)) ).

% vebt_minti_h
tff(fact_7062_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_maxti(Tia),aa_VEB7611521244314475576t_assn(aTP_Lamm_vp(T),Tia)) ).

% vebt_maxti_h
tff(fact_7063_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
    <=> ? [N5: nat] : ( Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N5))) ) ) ).

% zless_iff_Suc_zadd
tff(fact_7064_int__Suc,axiom,
    ! [Na: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) ) ).

% int_Suc
tff(fact_7065_int__ops_I4_J,axiom,
    ! [A: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ) ).

% int_ops(4)
tff(fact_7066_negative__zless__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7067_negD,axiom,
    ! [Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),zero_zero_int)
     => ? [N: nat] : ( Xa = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ) ).

% negD
tff(fact_7068_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))) ).

% not_zle_0_negative
tff(fact_7069_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : ( Z != aa_nat_int(semiri1314217659103216013at_int,N) )
     => ~ ! [N: nat] : ( Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ) ).

% int_cases
tff(fact_7070_int__of__nat__induct,axiom,
    ! [P2: fun_int_o,Z: int] :
      ( ! [N: nat] : aa_int_o(P2,aa_nat_int(semiri1314217659103216013at_int,N))
     => ( ! [N: nat] : aa_int_o(P2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
       => aa_int_o(P2,Z) ) ) ).

% int_of_nat_induct
tff(fact_7071_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_7072_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7073_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_7074_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% zle_int
tff(fact_7075_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_7076_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).

% int_ops(6)
tff(fact_7077_nat__less__as__int,axiom,
    ! [X3: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_less_as_int
tff(fact_7078_nat__leq__as__int,axiom,
    ! [X3: nat,Xa2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa2)) ) ).

% nat_leq_as_int
tff(fact_7079_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_7080_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_7081_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_7082_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : ( M != aa_nat_int(semiri1314217659103216013at_int,N) )
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_7083_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_7084_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_7085_zdiff__int__split,axiom,
    ! [P2: fun_int_o,Xa: nat,Y: nat] :
      ( aa_int_o(P2,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
         => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xa)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y)
         => aa_int_o(P2,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_7086_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,vEBT_Leaf((A),(B))) = one_one_nat ) ).

% VEBT_internal.cnt'.simps(1)
tff(fact_7087_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_bs,aTP_Lamm_br),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_7088_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vq(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_7089_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_VEBT_minNulli(Tia),aa_VEB6687300313434771872o_assn(aTP_Lamm_vl(T),Tia)),one_one_nat) ).

% minNrulli_ruleT
tff(fact_7090_T__vebt__buildupi,axiom,
    ! [Na: nat,H2: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(vEBT_V739175172307565963ildupi(Na),H2)),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% T_vebt_buildupi
tff(fact_7091_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_maxti(Tia),aa_VEB7611521244314475576t_assn(aTP_Lamm_vp(T),Tia)),one_one_nat) ).

% vebt_maxti_hT
tff(fact_7092_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_minti(Tia),aa_VEB7611521244314475576t_assn(aTP_Lamm_vo(T),Tia)),one_one_nat) ).

% vebt_minti_hT
tff(fact_7093_htt__vebt__buildupi_H,axiom,
    ! [Na: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% htt_vebt_buildupi'
tff(fact_7094_htt__vebt__buildupi,axiom,
    ! [Na: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% htt_vebt_buildupi
tff(fact_7095_finite__interval__int1,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_vr(A),B))) ).

% finite_interval_int1
tff(fact_7096_finite__interval__int4,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_vs(A),B))) ).

% finite_interval_int4
tff(fact_7097_finite__interval__int3,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_vt(A),B))) ).

% finite_interval_int3
tff(fact_7098_finite__interval__int2,axiom,
    ! [A: int,B: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_vu(A),B))) ).

% finite_interval_int2
tff(fact_7099_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_7100_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% zle_add1_eq_le
tff(fact_7101_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% zle_diff1_eq
tff(fact_7102_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N: nat] : ( K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% nonpos_int_cases
tff(fact_7103_negative__zle__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),zero_zero_int) ).

% negative_zle_0
tff(fact_7104_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N: nat] : ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ).

% zero_le_imp_eq_int
tff(fact_7105_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N: nat] : ( K != aa_nat_int(semiri1314217659103216013at_int,N) ) ) ).

% nonneg_int_cases
tff(fact_7106_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z)
    <=> ? [N5: nat] : ( Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N5)) ) ) ).

% zle_iff_zadd
tff(fact_7107_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_7108_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Na) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( Na = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_7109_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_7110_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_7111_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_7112_int__gr__induct,axiom,
    ! [K: int,I: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_gr_induct
tff(fact_7113_int__less__induct,axiom,
    ! [I: int,K: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_less_induct
tff(fact_7114_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_7115_plusinfinity,axiom,
    ! [D: int,P4: fun_int_o,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P4,X)
          <=> aa_int_o(P4,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
             => ( aa_int_o(P2,X)
              <=> aa_int_o(P4,X) ) )
         => ( ? [X_13: int] : aa_int_o(P4,X_13)
           => ? [X_1: int] : aa_int_o(P2,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_7116_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P1,X)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
             => ( aa_int_o(P2,X)
              <=> aa_int_o(P1,X) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(P2,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_7117_verit__la__generic,axiom,
    ! [A: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xa)
      | ( A = Xa )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),A) ) ).

% verit_la_generic
tff(fact_7118_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
     => aa_int_o(aa_int_fun_int_o(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_7119_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% add1_zle_eq
tff(fact_7120_int__ge__induct,axiom,
    ! [K: int,I: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P2,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_ge_induct
tff(fact_7121_int__le__induct,axiom,
    ! [I: int,K: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P2,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_le_induct
tff(fact_7122_int__induct,axiom,
    ! [P2: fun_int_o,K: int,I: int] :
      ( aa_int_o(P2,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(P2,I2)
             => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P2,I2)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P2,I) ) ) ) ).

% int_induct
tff(fact_7123_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q3: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_7124_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q3: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_7125_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R: int,B2: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_7126_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R: int,B2: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_7127_q__pos__lemma,axiom,
    ! [B2: int,Q6: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q6)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_7128_incr__mult__lemma,axiom,
    ! [D: int,P2: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X: int] :
            ( aa_int_o(P2,X)
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P2,X3)
             => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_7129_pos__mult__pos__ge,axiom,
    ! [Xa: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Na),one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,Na),Xa)) ) ) ).

% pos_mult_pos_ge
tff(fact_7130_decr__mult__lemma,axiom,
    ! [D: int,P2: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X: int] :
            ( aa_int_o(P2,X)
           => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P2,X3)
             => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_7131_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_7132_less__1__helper,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),one_one_int)),M) ) ).

% less_1_helper
tff(fact_7133_imp__le__cong,axiom,
    ! [Xa: int,X6: int,P2: $o,P4: $o] :
      ( ( Xa = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P2)
          <=> (P4) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
           => (P2) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
           => (P4) ) ) ) ) ).

% imp_le_cong
tff(fact_7134_conj__le__cong,axiom,
    ! [Xa: int,X6: int,P2: $o,P4: $o] :
      ( ( Xa = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P2)
          <=> (P4) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
            & (P2) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
            & (P4) ) ) ) ) ).

% conj_le_cong
tff(fact_7135_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_7136_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_7137_Gcd__int__greater__eq__0,axiom,
    ! [K4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K4)) ).

% Gcd_int_greater_eq_0
tff(fact_7138_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_7139_Tbuildupi__buildupi_H,axiom,
    ! [Na: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) = aa_nat_int(vEBT_V9176841429113362141ildupi,Na) ) ).

% Tbuildupi_buildupi'
tff(fact_7140_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,zero_zero_nat) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_7141_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_7142_finite__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_vv(Na),C2))) ) ).

% finite_nth_roots
tff(fact_7143_foldr__same,axiom,
    ! [Xs: list_real,Y: real] :
      ( ! [X: real,Y3: real] :
          ( aa_set_real_o(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs))
         => ( aa_set_real_o(member_real(Y3),aa_lis3769085826154824284t_real(set_real2,Xs))
           => ( X = Y3 ) ) )
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),aa_lis3769085826154824284t_real(set_real2,Xs))
           => ( X = Y ) )
       => ( aa_real_real(foldr_real_real(plus_plus_real,Xs),zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_7144_vebt__maxti__rule,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi,Y: nat] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_maxti(Tia),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aa_set5142912993642680584t_assn(aTP_Lamm_vw(Na),Sa),Tia),Y)),one_one_nat) ).

% vebt_maxti_rule
tff(fact_7145_foldr0,axiom,
    ! [Xs: list_real,C2: real,D: real] : ( aa_real_real(foldr_real_real(plus_plus_real,Xs),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs),D)),C2) ) ).

% foldr0
tff(fact_7146_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList)),zero_zero_nat)) ) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_7147_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_7148_card__nth__roots,axiom,
    ! [C2: complex,Na: nat] :
      ( ( C2 != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_vx(C2),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_7149_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_vy,Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_7150_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_vz,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_vy,Na))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_7151_sum__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_vz,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_vv(Na),C2))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_7152_vebt__buildupi__rule__basic,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot_set_nat)) ) ).

% vebt_buildupi_rule_basic
tff(fact_7153_vebt__minti__rule,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi,Y: nat] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_minti(Tia),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aa_set5142912993642680584t_assn(aTP_Lamm_wa(Na),Sa),Tia),Y)),one_one_nat) ).

% vebt_minti_rule
tff(fact_7154_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = one_one_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),vEBT_Leaf((A3),(B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_7155_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList)),zero_zero_real)) ) ).

% VEBT_internal.cnt.simps(2)
tff(fact_7156_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary2))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_7157_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = one_one_real )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),vEBT_Leaf((A3),(B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary2))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_7158_prod__decode__aux_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xa),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xa)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_7159_vebt__buildupi__refines,axiom,
    ! [Na: nat] : refine5565527176597971370_VEBTi(vEBT_vebt_buildupi(Na),vEBT_V739175172307565963ildupi(Na)) ).

% vebt_buildupi_refines
tff(fact_7160_vebt__deletei__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine5565527176597971370_VEBTi(vEBT_vebt_deletei(Tia,Xa),vEBT_V1365221501068881998eletei(T,Tia,Xa)) ).

% vebt_deletei_refines
tff(fact_7161_vebt__inserti__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine5565527176597971370_VEBTi(vEBT_vebt_inserti(Tia,Xa),vEBT_V3964819847710782039nserti(T,Tia,Xa)) ).

% vebt_inserti_refines
tff(fact_7162_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : ( aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,vEBT_Nodei(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(size_size_VEBT_VEBTi,X13)),aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ) ).

% VEBTi.size(3)
tff(fact_7163_vebt__succi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine7594492741263601813on_nat(vEBT_vebt_succi(Tia,Xa),vEBT_VEBT_vebt_succi(T,Tia,Xa)) ).

% vebt_succi_refines
tff(fact_7164_vebt__predi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine7594492741263601813on_nat(vEBT_vebt_predi(Tia,Xa),vEBT_VEBT_vebt_predi(T,Tia,Xa)) ).

% vebt_predi_refines
tff(fact_7165_vebt__memberi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T: vEBT_VEBT] : refine_Imp_refines_o(vEBT_vebt_memberi(Tia,Xa),vEBT_V854960066525838166emberi(T,Tia,Xa)) ).

% vebt_memberi_refines
tff(fact_7166_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : ( aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,vEBT_Nodei(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(vEBT_size_VEBTi,X13)),aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ) ).

% VEBTi.size_gen(1)
tff(fact_7167_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
     => aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(T,Tia,Xa)),one_one_nat) ) ).

% TBOUND_minNull
tff(fact_7168_TBOUND__vebt__buildupi,axiom,
    ! [Na: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V739175172307565963ildupi(Na)),aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) ).

% TBOUND_vebt_buildupi
tff(fact_7169_finite__atMost,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% finite_atMost
tff(fact_7170_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_7171_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat) ).

% atMost_0
tff(fact_7172_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_7173_atMost__Suc,axiom,
    ! [K: nat] : ( aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(suc,K)),aa_nat_set_nat(set_ord_atMost_nat,K)) ) ).

% atMost_Suc
tff(fact_7174_finite__nat__iff__bounded__le,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_atMost_nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_7175_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_atMost_nat,Na))) ) ).

% atMost_Suc_eq_insert_0
tff(fact_7176_polynomial__product__nat,axiom,
    ! [M: nat,A: fun_nat_nat,Na: nat,B: fun_nat_nat,Xa: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_nat(B,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_wb(A),Xa)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wb(B),Xa)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_wd(A),B),Xa)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_7177_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_vebt_minti(T)),one_one_nat) ).

% TBOUND_vebt_minti
tff(fact_7178_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_vebt_maxti(T)),one_one_nat) ).

% TBOUND_vebt_maxti
tff(fact_7179_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : aa_nat_o(time_TBOUND_o(vEBT_VEBT_minNulli(T)),one_one_nat) ).

% TBOUND_minNulli
tff(fact_7180_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) ).

% finite_atLeastAtMost
tff(fact_7181_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : ( aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ) ).

% image_Suc_atLeastAtMost
tff(fact_7182_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ) ).

% card_atLeastAtMost
tff(fact_7183_all__nat__less,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Na)
         => aa_nat_o(P2,M5) )
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))
         => aa_nat_o(P2,X5) ) ) ).

% all_nat_less
tff(fact_7184_ex__nat__less,axiom,
    ! [Na: nat,P2: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Na)
          & aa_nat_o(P2,M5) )
    <=> ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))
          & aa_nat_o(P2,X5) ) ) ).

% ex_nat_less
tff(fact_7185_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(L),aa_nat_nat(suc,U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_7186_atMost__atLeast0,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_ord_atMost_nat,Na) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na) ) ).

% atMost_atLeast0
tff(fact_7187_complex__mod__minus__le__complex__mod,axiom,
    ! [Xa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Xa))),aa_complex_real(real_V1022390504157884413omplex,Xa)) ).

% complex_mod_minus_le_complex_mod
tff(fact_7188_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% complex_mod_triangle_ineq2
tff(fact_7189_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(suc,Na)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) ) ).

% atLeast0_atMost_Suc
tff(fact_7190_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na) = aa_set_nat_set_nat(insert_nat(M),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_7191_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Na)) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(suc,Na)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_7192_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_set_nat(insert_nat(M),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Na)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_7193_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N6: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N6),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))
     => aa_set_nat_o(finite_finite_nat,N6) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_7194_image__Suc__lessThan,axiom,
    ! [Na: nat] : ( aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Na) ) ).

% image_Suc_lessThan
tff(fact_7195_image__Suc__atMost,axiom,
    ! [Na: nat] : ( aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,Na)) ) ).

% image_Suc_atMost
tff(fact_7196_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Na)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na))) ) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_7197_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Na),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat)) ) ).

% atLeast1_atMost_eq_remove0
tff(fact_7198_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na)) = aa_set_int_set_int(insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na)),set_or1266510415728281911st_int(M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_7199_simp__from__to,axiom,
    ! [I: int,J: int] :
      ( set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,aa_set_int_set_int(insert_int(I),set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J))) ) ).

% simp_from_to
tff(fact_7200_aset_I7_J,axiom,
    ! [D4: int,A4: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ).

% aset(7)
tff(fact_7201_aset_I5_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(T),A4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),T) ) ) ) ) ).

% aset(5)
tff(fact_7202_aset_I4_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(T),A4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( X3 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4) != T ) ) ) ) ) ).

% aset(4)
tff(fact_7203_aset_I3_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( ( X3 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4) = T ) ) ) ) ) ).

% aset(3)
tff(fact_7204_bset_I7_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(T),B4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ).

% bset(7)
tff(fact_7205_bset_I5_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),T) ) ) ) ).

% bset(5)
tff(fact_7206_bset_I4_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(T),B4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( X3 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4) != T ) ) ) ) ) ).

% bset(4)
tff(fact_7207_bset_I3_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( ( X3 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4) = T ) ) ) ) ) ).

% bset(3)
tff(fact_7208_periodic__finite__ex,axiom,
    ! [D: int,P2: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P2,X)
          <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [X_12: int] : aa_int_o(P2,X_12)
        <=> ? [X5: int] :
              ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P2,X5) ) ) ) ) ).

% periodic_finite_ex
tff(fact_7209_aset_I8_J,axiom,
    ! [D4: int,A4: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ).

% aset(8)
tff(fact_7210_aset_I6_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),T) ) ) ) ) ).

% aset(6)
tff(fact_7211_bset_I8_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B4)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ).

% bset(8)
tff(fact_7212_bset_I6_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( aa_set_int_o(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),T) ) ) ) ).

% bset(6)
tff(fact_7213_cpmi,axiom,
    ! [D4: int,P2: fun_int_o,P4: fun_int_o,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
           => ( aa_int_o(P2,X)
            <=> aa_int_o(P4,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),B4)
                     => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
             => ( aa_int_o(P2,X)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)) ) )
         => ( ! [X: int,K2: int] :
                ( aa_int_o(P4,X)
              <=> aa_int_o(P4,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
           => ( ? [X_12: int] : aa_int_o(P2,X_12)
            <=> ( ? [X5: int] :
                    ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P4,X5) )
                | ? [X5: int] :
                    ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B4)
                        & aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X5)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_7214_cppi,axiom,
    ! [D4: int,P2: fun_int_o,P4: fun_int_o,A4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
           => ( aa_int_o(P2,X)
            <=> aa_int_o(P4,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),A4)
                     => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
             => ( aa_int_o(P2,X)
               => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)) ) )
         => ( ! [X: int,K2: int] :
                ( aa_int_o(P4,X)
              <=> aa_int_o(P4,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
           => ( ? [X_12: int] : aa_int_o(P2,X_12)
            <=> ( ? [X5: int] :
                    ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P4,X5) )
                | ? [X5: int] :
                    ( aa_set_int_o(member_int(X5),set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A4)
                        & aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X5)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_7215_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) ) ).

% nat_mult_div_cancel_disj
tff(fact_7216_zdiv__int,axiom,
    ! [A: nat,B: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% zdiv_int
tff(fact_7217_real__of__nat__div4,axiom,
    ! [Na: nat,Xa: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xa))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))) ).

% real_of_nat_div4
tff(fact_7218_real__of__nat__div2,axiom,
    ! [Na: nat,Xa: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xa)))) ).

% real_of_nat_div2
tff(fact_7219_real__of__nat__div3,axiom,
    ! [Na: nat,Xa: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Xa))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xa)))),one_one_real) ).

% real_of_nat_div3
tff(fact_7220_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_7221_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_7222_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_7223_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_7224_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_7225_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_7226_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_7227_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_7228_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_7229_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_7230_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)) ) ) ).

% div_positive_int
tff(fact_7231_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_7232_zdiv__mono2__neg,axiom,
    ! [A: int,B2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_7233_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_7234_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_7235_zdiv__mono2,axiom,
    ! [A: int,B2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ) ).

% zdiv_mono2
tff(fact_7236_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B)) ) ) ).

% zdiv_mono1
tff(fact_7237_int__div__less__self,axiom,
    ! [Xa: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),K)),Xa) ) ) ).

% int_div_less_self
tff(fact_7238_div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = 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),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_7239_msrevs_I1_J,axiom,
    ! [Na: nat,K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),K) ) ) ).

% msrevs(1)
tff(fact_7240_verit__less__mono__div__int2,axiom,
    ! [A4: int,B4: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Na))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_7241_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_7242_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_7243_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_7244_split__zdiv,axiom,
    ! [P2: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P2,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P2,I3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P2,I3) ) ) ) ) ).

% split_zdiv
tff(fact_7245_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(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_7246_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_7247_int__div__minus__is__minus1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = aa_int_int(uminus_uminus_int,A) )
      <=> ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_7248_int__div__same__is__1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = A )
      <=> ( B = one_one_int ) ) ) ).

% int_div_same_is_1
tff(fact_7249_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_7250_div__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = zero_zero_nat ) ) ).

% div_less
tff(fact_7251_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = M )
      <=> ( Na = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_7252_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),Na) = M ) ) ).

% div_mult_self1_is_m
tff(fact_7253_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),Na) = M ) ) ).

% div_mult_self_is_m
tff(fact_7254_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_7255_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_7256_div__le__mono,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),K)) ) ).

% div_le_mono
tff(fact_7257_div__le__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),M) ).

% div_le_dividend
tff(fact_7258_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( Na = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_7259_Suc__div__le__mono,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),Na)) ).

% Suc_div_le_mono
tff(fact_7260_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_7261_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))),M) ).

% times_div_less_eq_dividend
tff(fact_7262_div__times__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na)),M) ).

% div_times_less_eq_dividend
tff(fact_7263_div__mult__le,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),A) ).

% div_mult_le
tff(fact_7264_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% div_greater_zero_iff
tff(fact_7265_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M)) ) ) ).

% div_le_mono2
tff(fact_7266_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),M) ) ) ).

% div_less_dividend
tff(fact_7267_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q3)),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_7268_td__gal__lt,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)),B) ) ) ).

% td_gal_lt
tff(fact_7269_zdiv__le__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),A) ) ) ).

% zdiv_le_dividend
tff(fact_7270_zdiv__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_7271_div__if,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = $ite(
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
          | ( Na = zero_zero_nat ) ),
          zero_zero_nat,
          aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ) ) ).

% div_if
tff(fact_7272_div__nat__eqI,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q3)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_7273_split__div,axiom,
    ! [P2: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P2,zero_zero_nat) )
        & ( ( Na != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I3)),J3) )
               => aa_nat_o(P2,I3) ) ) ) ) ) ).

% split_div
tff(fact_7274_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_7275_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)))) ) ).

% dividend_less_times_div
tff(fact_7276_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Q3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_7277_td__gal,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C2)) ) ) ).

% td_gal
tff(fact_7278_divide__nat__def,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = $ite(Na = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_we(M),Na)))) ) ).

% divide_nat_def
tff(fact_7279_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = 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),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_7280_split__div_H,axiom,
    ! [P2: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
          & aa_nat_o(P2,zero_zero_nat) )
        | ? [Q7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q7)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q7)))
            & aa_nat_o(P2,Q7) ) ) ) ).

% split_div'
tff(fact_7281_power__sub,axiom,
    ! [Na: nat,M: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Na)) ) ) ) ).

% power_sub
tff(fact_7282_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_7283_nat__mod__eq_H,axiom,
    ! [A: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Na) = A ) ) ).

% nat_mod_eq'
tff(fact_7284_mod__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = M ) ) ).

% mod_less
tff(fact_7285_binomial__Suc__n,axiom,
    ! [Na: nat] : ( aa_nat_nat(binomial(aa_nat_nat(suc,Na)),Na) = aa_nat_nat(suc,Na) ) ).

% binomial_Suc_n
tff(fact_7286_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_7287_binomial__1,axiom,
    ! [Na: nat] : ( aa_nat_nat(binomial(Na),aa_nat_nat(suc,zero_zero_nat)) = Na ) ).

% binomial_1
tff(fact_7288_binomial__0__Suc,axiom,
    ! [K: nat] : ( aa_nat_nat(binomial(zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ) ).

% binomial_0_Suc
tff(fact_7289_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_nat(binomial(Na),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% binomial_eq_0_iff
tff(fact_7290_binomial__Suc__Suc,axiom,
    ! [Na: nat,K: nat] : ( aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),aa_nat_nat(suc,K))) ) ).

% binomial_Suc_Suc
tff(fact_7291_binomial__n__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(binomial(Na),zero_zero_nat) = one_one_nat ) ).

% binomial_n_0
tff(fact_7292_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ) ).

% Suc_mod_mult_self1
tff(fact_7293_Suc__mod__mult__self2,axiom,
    ! [M: nat,Na: nat,K: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ) ).

% Suc_mod_mult_self2
tff(fact_7294_Suc__mod__mult__self3,axiom,
    ! [K: nat,Na: nat,M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ) ).

% Suc_mod_mult_self3
tff(fact_7295_Suc__mod__mult__self4,axiom,
    ! [Na: nat,K: nat,M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)),M))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ) ).

% Suc_mod_mult_self4
tff(fact_7296_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(binomial(Na),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% zero_less_binomial_iff
tff(fact_7297_mod__Suc__eq,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ) ).

% mod_Suc_eq
tff(fact_7298_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),Na) ) ).

% mod_Suc_Suc_eq
tff(fact_7299_nat__mod__eq,axiom,
    ! [B: nat,Na: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Na)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Na) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Na) = B ) ) ) ).

% nat_mod_eq
tff(fact_7300_mod__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),M) ).

% mod_less_eq_dividend
tff(fact_7301_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_nat(binomial(Na),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_7302_Suc__times__binomial,axiom,
    ! [K: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(binomial(Na),K)) ) ).

% Suc_times_binomial
tff(fact_7303_Suc__times__binomial__eq,axiom,
    ! [Na: nat,K: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(binomial(Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ) ).

% Suc_times_binomial_eq
tff(fact_7304_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(binomial(Na),K) = aa_nat_nat(binomial(Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% binomial_symmetric
tff(fact_7305_binomial__le__pow,axiom,
    ! [R: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),R)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R)) ) ).

% binomial_le_pow
tff(fact_7306_mod__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = Na,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))) ) ).

% mod_Suc
tff(fact_7307_mod__induct,axiom,
    ! [P2: fun_nat_o,Na: nat,P3: nat,M: nat] :
      ( aa_nat_o(P2,Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),P3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P3)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P3)
               => ( aa_nat_o(P2,N)
                 => aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P3)) ) )
           => aa_nat_o(P2,M) ) ) ) ) ).

% mod_induct
tff(fact_7308_gcd__nat__induct,axiom,
    ! [P2: fun_nat_fun_nat_o,M: nat,Na: nat] :
      ( ! [M3: nat] : aa_nat_o(aa_nat_fun_nat_o(P2,M3),zero_zero_nat)
     => ( ! [M3: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( aa_nat_o(aa_nat_fun_nat_o(P2,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M3),N))
             => aa_nat_o(aa_nat_fun_nat_o(P2,M3),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P2,M),Na) ) ) ).

% gcd_nat_induct
tff(fact_7309_nat__mod__lem,axiom,
    ! [Na: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),Na)
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Na) = B ) ) ) ).

% nat_mod_lem
tff(fact_7310_mod__less__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),Na) ) ).

% mod_less_divisor
tff(fact_7311_mod__Suc__le__divisor,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_7312_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),L)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),L)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),Na)),L) = Na ) ) ) ).

% word_rot_lem
tff(fact_7313_nat__minus__mod,axiom,
    ! [Na: nat,M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),M))),M) = zero_zero_nat ) ).

% nat_minus_mod
tff(fact_7314_mod__geq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) ) ) ).

% mod_geq
tff(fact_7315_mod__if,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ) ).

% mod_if
tff(fact_7316_mod__nat__sub,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),Y) ) ) ).

% mod_nat_sub
tff(fact_7317_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q5: nat] : ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q5) ) ) ).

% mod_eq_0D
tff(fact_7318_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_7319_nat__mod__eq__iff,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Na) )
    <=> ? [Q1: nat,Q22: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q22)) ) ) ).

% nat_mod_eq_iff
tff(fact_7320_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(binomial(Na),K)) ) ).

% zero_less_binomial
tff(fact_7321_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(binomial(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(binomial(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ) ).

% Suc_times_binomial_add
tff(fact_7322_binomial__Suc__Suc__eq__times,axiom,
    ! [Na: nat,K: nat] : ( aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(binomial(Na),K))),aa_nat_nat(suc,K)) ) ).

% binomial_Suc_Suc_eq_times
tff(fact_7323_choose__mult,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(Na),M)),aa_nat_nat(binomial(M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_7324_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),Na) ) ).

% mod_le_divisor
tff(fact_7325_div__less__mono,axiom,
    ! [A4: nat,B4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),Na) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Na) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_7326_mod__nat__add,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y)),Z) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y)),Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_7327_nat__mod__eq__lemma,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xa)
       => ? [Q5: nat] : ( Xa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q5)) ) ) ) ).

% nat_mod_eq_lemma
tff(fact_7328_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,Na: 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,Na),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ~ ! [S3: nat] : ( Na != 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_7329_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,Na: 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,Na),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ~ ! [S3: nat] : ( M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ) ).

% mod_eq_nat1E
tff(fact_7330_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_wf(M)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,M)) ) ).

% sum_choose_upper
tff(fact_7331_binomial__absorption,axiom,
    ! [K: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(binomial(Na),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ) ).

% binomial_absorption
tff(fact_7332_split__mod,axiom,
    ! [P2: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P2,M) )
        & ( ( Na != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I3)),J3) )
               => aa_nat_o(P2,J3) ) ) ) ) ) ).

% split_mod
tff(fact_7333_mod__lemma,axiom,
    ! [C2: nat,R: nat,B: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Q3),C2))),R)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C2)) ) ) ).

% mod_lemma
tff(fact_7334_sum__choose__lower,axiom,
    ! [R: nat,Na: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_wg(R)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(binomial(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R),Na))),Na) ) ).

% sum_choose_lower
tff(fact_7335_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(binomial(Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_7336_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(binomial(Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_7337_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_7338_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wh(M),Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(binomial(aa_nat_nat(suc,Na)),M) ) ) ).

% sum_choose_diagonal
tff(fact_7339_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(binomial(Na),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_7340_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),Na)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Na) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Na)) ) ) ).

% verit_le_mono_div
tff(fact_7341_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo: option4927543243414619207at_nat,Deg: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
      ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg,Tree_list),Summary)),vEBT_Nodei(Mmoi,Degi,Tree_array,Summaryi)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
            aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                aa_o_assn(pure_assn,
                  ( ( Mmoi = Mmo )
                  & ( Degi = Deg ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_wi(Tree_list),Tree_array))) ) ).

% vebt_assn_raw.simps(2)
tff(fact_7342_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      ( upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ) ).

% upto_aux_rec
tff(fact_7343_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_7344_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_7345_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_7346_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% neg_mod_bound
tff(fact_7347_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_7348_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_7349_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_7350_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
    <=> ? [Q7: int] : ( M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q7) ) ) ).

% zmod_eq_0_iff
tff(fact_7351_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
     => ? [Q5: int] : ( M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q5) ) ) ).

% zmod_eq_0D
tff(fact_7352_zmod__int,axiom,
    ! [A: nat,B: nat] : ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% zmod_int
tff(fact_7353_int__mod__eq,axiom,
    ! [B: int,Na: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),Na)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Na) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na) )
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Na) = B ) ) ) ) ).

% int_mod_eq
tff(fact_7354_int__mod__lem,axiom,
    ! [Na: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),Na) )
      <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na) = B ) ) ) ).

% int_mod_lem
tff(fact_7355_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_7356_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_7357_int__mod__ge,axiom,
    ! [A: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),Na)) ) ) ).

% int_mod_ge
tff(fact_7358_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% neg_mod_conj
tff(fact_7359_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_7360_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_7361_int__mod__le_H,axiom,
    ! [B: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Na))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),Na)) ) ).

% int_mod_le'
tff(fact_7362_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ) ).

% zmod_zminus2_eq_if
tff(fact_7363_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ) ).

% zmod_zminus1_eq_if
tff(fact_7364_nonneg__mod__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% nonneg_mod_div
tff(fact_7365_zdiv__mono__strict,axiom,
    ! [A4: int,B4: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),Na) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Na) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_7366_int__mod__ge_H,axiom,
    ! [B: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),Na)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),Na)) ) ) ).

% int_mod_ge'
tff(fact_7367_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_7368_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_7369_mod__power__lem,axiom,
    ! [A: int,Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A),Na)),aa_nat_int(power_power_int(A),M)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),zero_zero_int,aa_nat_int(power_power_int(A),Na)) ) ) ).

% mod_power_lem
tff(fact_7370_split__zmod,axiom,
    ! [P2: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P2,Na) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P2,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P2,J3) ) ) ) ) ).

% split_zmod
tff(fact_7371_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_7372_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q3)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_7373_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_7374_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_7375_mod__add__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_7376_mod__sub__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xa),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_7377_zmod__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_7378_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_7379_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_7380_verit__le__mono__div__int,axiom,
    ! [A4: int,B4: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),Na)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Na) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_7381_split__neg__lemma,axiom,
    ! [K: int,P2: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P2,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_7382_split__pos__lemma,axiom,
    ! [K: int,P2: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P2,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_7383_vebt__assn__raw_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xa),Xaa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
               => ( Y != aa_o_assn(pure_assn,
                      ( ( (Ai) = (A3) )
                      & ( (Bi) = (B3) ) )) ) ) )
       => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary2) )
             => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                  ( ( Xaa = vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                          aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                              aa_o_assn(pure_assn,
                                ( ( Mmoi2 = Mmo2 )
                                & ( Degi2 = Deg2 ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_wi(Tree_list2),Tree_array2))) ) ) )
         => ( ( ? [V: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Va,Vb3),Vc3) )
             => ( ? [Vd3: $o,Ve3: $o] : ( Xaa = vEBT_Leafi((Vd3),(Ve3)) )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] : ( Xa = vEBT_Leaf((Vd3),(Ve3)) )
               => ( ? [V: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : ( Xaa = vEBT_Nodei(V,Va,Vb3,Vc3) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_7384_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(Xa),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xa))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7385_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(Xa),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,Xa)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7386_VEBTi_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leafi((X21),(X22)) = vEBT_Leafi((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBTi.inject(2)
tff(fact_7387_exp__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y)) ) ).

% exp_less_mono
tff(fact_7388_exp__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% exp_less_cancel_iff
tff(fact_7389_exp__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% exp_le_cancel_iff
tff(fact_7390_one__less__exp__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% one_less_exp_iff
tff(fact_7391_exp__less__one__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_7392_one__le__exp__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% one_le_exp_iff
tff(fact_7393_exp__le__one__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_7394_exp__less__cancel,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),aa_real_real(exp_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% exp_less_cancel
tff(fact_7395_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,X21: $o,X22: $o] : ( vEBT_Nodei(X11,X12,X13,X14) != vEBT_Leafi((X21),(X22)) ) ).

% VEBTi.distinct(1)
tff(fact_7396_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: array_VEBT_VEBTi,X142: vEBT_VEBTi] : ( Y != vEBT_Nodei(X112,X122,X132,X142) )
     => ~ ! [X212: $o,X222: $o] : ( Y != vEBT_Leafi((X212),(X222)) ) ) ).

% VEBTi.exhaust
tff(fact_7397_not__exp__less__zero,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xa)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_7398_exp__gt__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xa)) ).

% exp_gt_zero
tff(fact_7399_exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X: real] : ( aa_real_real(exp_real,X) = Y ) ) ).

% exp_total
tff(fact_7400_exp__ge__zero,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xa)) ).

% exp_ge_zero
tff(fact_7401_not__exp__le__zero,axiom,
    ! [Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_7402_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : ( aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,vEBT_Leafi((X21),(X22))) = zero_zero_nat ) ).

% VEBTi.size(4)
tff(fact_7403_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : ( aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,vEBT_Leafi((X21),(X22))) = zero_zero_nat ) ).

% VEBTi.size_gen(2)
tff(fact_7404_exp__gt__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xa)) ) ).

% exp_gt_one
tff(fact_7405_exp__ge__add__one__self,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(exp_real,Xa)) ).

% exp_ge_add_one_self
tff(fact_7406_vebt__assn__raw_Ocases,axiom,
    ! [Xa: produc3625547720036274456_VEBTi] :
      ( ! [A3: $o,B3: $o,Ai: $o,Bi: $o] : ( Xa != produc6084888613844515218_VEBTi(vEBT_Leaf((A3),(B3)),vEBT_Leafi((Ai),(Bi))) )
     => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] : ( Xa != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary2),vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2)) )
       => ( ! [V: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] : ( Xa != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Va,Vb3),Vc3),vEBT_Leafi((Vd3),(Ve3))) )
         => ~ ! [Vd3: $o,Ve3: $o,V: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : ( Xa != produc6084888613844515218_VEBTi(vEBT_Leaf((Vd3),(Ve3)),vEBT_Nodei(V,Va,Vb3,Vc3)) ) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_7407_VEBT__internal_OminNulli_Ocases,axiom,
    ! [Xa: vEBT_VEBTi] :
      ( ( Xa != vEBT_Leafi($false,$false) )
     => ( ! [Uv: $o] : ( Xa != vEBT_Leafi($true,(Uv)) )
       => ( ! [Uu: $o] : ( Xa != vEBT_Leafi((Uu),$true) )
         => ( ! [Uw: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] : ( Xa != vEBT_Nodei(none_P5556105721700978146at_nat,Uw,Ux,Uy) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : ( Xa != vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ) ).

% VEBT_internal.minNulli.cases
tff(fact_7408_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Ai2: $o,Bi2: $o] :
      ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_Leaf((A),(B))),vEBT_Leafi((Ai2),(Bi2))) = aa_o_assn(pure_assn,
          ( ( (Ai2) = (A) )
          & ( (Bi2) = (B) ) )) ) ).

% vebt_assn_raw.simps(1)
tff(fact_7409_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V2: option4927543243414619207at_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Vd2: $o,Ve2: $o] : ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb),Vc)),vEBT_Leafi((Vd2),(Ve2))) = bot_bot_assn ) ).

% vebt_assn_raw.simps(3)
tff(fact_7410_exp__ge__add__one__self__aux,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(exp_real,Xa)) ) ).

% exp_ge_add_one_self_aux
tff(fact_7411_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(minus_minus_real(Y),one_one_real))
          & ( aa_real_real(exp_real,X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_7412_vebt__minti_Ocases,axiom,
    ! [Xa: vEBT_VEBTi] :
      ( ! [A3: $o,B3: $o] : ( Xa != vEBT_Leafi((A3),(B3)) )
     => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : ( Xa != vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : ( Xa != vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2) ) ) ) ).

% vebt_minti.cases
tff(fact_7413_vebt__assn__def,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia) = ex_assn_VEBT_VEBT(aa_VEB2970430004736285756T_assn(aa_set5977240410424667023T_assn(aTP_Lamm_wj(Na),Sa),Tia)) ) ).

% vebt_assn_def
tff(fact_7414_vebt__assn__raw_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xa),Xaa) = Y )
     => ( accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
                 => ( ( Y = aa_o_assn(pure_assn,
                          ( ( (Ai) = (A3) )
                          & ( (Bi) = (B3) ) )) )
                   => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf((A3),(B3)),vEBT_Leafi((Ai),(Bi)))) ) ) )
         => ( ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary2) )
               => ! [Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
                    ( ( Xaa = vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2) )
                   => ( ( Y = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                              aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                                  aa_o_assn(pure_assn,
                                    ( ( Mmoi2 = Mmo2 )
                                    & ( Degi2 = Deg2 ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_wi(Tree_list2),Tree_array2))) )
                     => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary2),vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2))) ) ) )
           => ( ! [V: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Va,Vb3),Vc3) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xaa = vEBT_Leafi((Vd3),(Ve3)) )
                     => ( ( Y = bot_bot_assn )
                       => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Va,Vb3),Vc3),vEBT_Leafi((Vd3),(Ve3)))) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( Xa = vEBT_Leaf((Vd3),(Ve3)) )
                   => ! [V: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xaa = vEBT_Nodei(V,Va,Vb3,Vc3) )
                       => ( ( Y = bot_bot_assn )
                         => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf((Vd3),(Ve3)),vEBT_Nodei(V,Va,Vb3,Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_7415_vebt__minti_Oelims,axiom,
    ! [Xa: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_minti(Xa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leafi((A3),(B3)) )
           => ( Y != $ite(
                  (A3),
                  heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),
                  $ite((B3),heap_T3487192422709364219on_nat(some_nat(one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : ( Xa = vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
           => ( Y != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : ( Xa = vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2) )
               => ( Y != heap_T3487192422709364219on_nat(some_nat(Mi2)) ) ) ) ) ) ).

% vebt_minti.elims
tff(fact_7416_vebt__maxti_Oelims,axiom,
    ! [Xa: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_maxti(Xa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leafi((A3),(B3)) )
           => ( Y != $ite(
                  (B3),
                  heap_T3487192422709364219on_nat(some_nat(one_one_nat)),
                  $ite((A3),heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : ( Xa = vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
           => ( Y != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : ( Xa = vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2) )
               => ( Y != heap_T3487192422709364219on_nat(some_nat(Ma2)) ) ) ) ) ) ).

% vebt_maxti.elims
tff(fact_7417_fact__ge__self,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_self
tff(fact_7418_fact__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono_nat
tff(fact_7419_fact__less__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_7420_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_7421_fact__div__fact__le__pow,axiom,
    ! [R: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),R)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R)) ) ).

% fact_div_fact_le_pow
tff(fact_7422_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( 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,Na),K)))),aa_nat_nat(binomial(Na),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_7423_prod__Suc__fact,axiom,
    ! [Na: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ) ).

% prod_Suc_fact
tff(fact_7424_prod__Suc__Suc__fact,axiom,
    ! [Na: nat] : ( aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ) ).

% prod_Suc_Suc_fact
tff(fact_7425_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : ( vEBT_vebt_minti(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = heap_T3487192422709364219on_nat(none_nat) ) ).

% vebt_minti.simps(2)
tff(fact_7426_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : ( vEBT_vebt_maxti(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) = heap_T3487192422709364219on_nat(none_nat) ) ).

% vebt_maxti.simps(2)
tff(fact_7427_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(binomial(Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),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,Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_7428_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : ( vEBT_vebt_minti(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz)) = heap_T3487192422709364219on_nat(some_nat(Mi)) ) ).

% vebt_minti.simps(3)
tff(fact_7429_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : ( vEBT_vebt_maxti(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz)) = heap_T3487192422709364219on_nat(some_nat(Ma)) ) ).

% vebt_maxti.simps(3)
tff(fact_7430_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_aq),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Na)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_7431_vebt__maxti_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( vEBT_vebt_maxti(vEBT_Leafi((A),(B))) = $ite(
          (B),
          heap_T3487192422709364219on_nat(some_nat(one_one_nat)),
          $ite((A),heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) ).

% vebt_maxti.simps(1)
tff(fact_7432_vebt__minti_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( vEBT_vebt_minti(vEBT_Leafi((A),(B))) = $ite(
          (A),
          heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),
          $ite((B),heap_T3487192422709364219on_nat(some_nat(one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) ).

% vebt_minti.simps(1)
tff(fact_7433_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_aq),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_7434_vebt__maxti_Opelims,axiom,
    ! [Xa: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_maxti(Xa) = Y )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leafi((A3),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      heap_T3487192422709364219on_nat(some_nat(one_one_nat)),
                      $ite((A3),heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) )
               => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),vEBT_Leafi((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( Xa = vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( Xa = vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = heap_T3487192422709364219on_nat(some_nat(Ma2)) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_maxti.pelims
tff(fact_7435_vebt__minti_Opelims,axiom,
    ! [Xa: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_minti(Xa) = Y )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leafi((A3),(B3)) )
             => ( ( Y = $ite(
                      (A3),
                      heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),
                      $ite((B3),heap_T3487192422709364219on_nat(some_nat(one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) )
               => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),vEBT_Leafi((A3),(B3))) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( Xa = vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( Xa = vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = heap_T3487192422709364219on_nat(some_nat(Mi2)) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_minti.pelims
tff(fact_7436_Maclaurin__lemma,axiom,
    ! [H2: real,F: fun_real_real,J: fun_nat_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
     => ? [B7: real] : ( aa_real_real(F,H2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_wk(H2),J),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B7),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,H2),Na)),aa_nat_real(semiri2265585572941072030t_real,Na)))) ) ) ).

% Maclaurin_lemma
tff(fact_7437_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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)),Na)) = 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)),Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_7438_sumr__cos__zero__one,axiom,
    ! [Na: nat] : ( groups6591440286371151544t_real(aTP_Lamm_wl,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = one_one_real ) ).

% sumr_cos_zero_one
tff(fact_7439_Maclaurin__exp__lt,axiom,
    ! [Xa: real,Na: nat] :
      ( ( Xa != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(exp_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_wm,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(exp_real,T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_7440_Maclaurin__exp__le,axiom,
    ! [Xa: real,Na: nat] :
    ? [T6: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(exp_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_wm,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(exp_real,T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_7441_Least__eq__0,axiom,
    ! [P2: fun_nat_o] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( ord_Least_nat(P2) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_7442_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_7443_artanh__minus__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xa)) ) ) ).

% artanh_minus_real
tff(fact_7444_Least__Suc2,axiom,
    ! [P2: fun_nat_o,Na: nat,Q2: fun_nat_o,M: nat] :
      ( aa_nat_o(P2,Na)
     => ( aa_nat_o(Q2,M)
       => ( ~ aa_nat_o(P2,zero_zero_nat)
         => ( ! [K2: nat] :
                ( aa_nat_o(P2,aa_nat_nat(suc,K2))
              <=> aa_nat_o(Q2,K2) )
           => ( ord_Least_nat(P2) = aa_nat_nat(suc,ord_Least_nat(Q2)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_7445_Least__Suc,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( aa_nat_o(P2,Na)
     => ( ~ aa_nat_o(P2,zero_zero_nat)
       => ( ord_Least_nat(P2) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_wn(P2))) ) ) ) ).

% Least_Suc
tff(fact_7446_abs__real__def,axiom,
    ! [A: real] :
      ( aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ) ).

% abs_real_def
tff(fact_7447_sin__bound__lemma,axiom,
    ! [Xa: real,Y: real,U: real,V2: real] :
      ( ( Xa = Y )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),U)),Y))),V2) ) ) ).

% sin_bound_lemma
tff(fact_7448_lemma__interval,axiom,
    ! [A: real,Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xa),Y4))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_7449_lemma__interval__lt,axiom,
    ! [A: real,Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xa),Y4))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_7450_set__vebt__succ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( vEBT_vebt_succ(T,Xa) = $ite(
            ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),vEBT_set_vebt(T))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X5) ),
            some_nat(ord_Least_nat(aa_nat_fun_nat_o(aTP_Lamm_wo(T),Xa))),
            none_nat ) ) ) ).

% set_vebt_succ'
tff(fact_7451_arctan__add,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)))) ) ) ) ).

% arctan_add
tff(fact_7452_cos__coeff__Suc,axiom,
    ! [Na: nat] : ( aa_nat_real(cos_coeff,aa_nat_nat(suc,Na)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Na))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ) ).

% cos_coeff_Suc
tff(fact_7453_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_7454_zero__less__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% zero_less_arctan_iff
tff(fact_7455_arctan__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_7456_zero__le__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% zero_le_arctan_iff
tff(fact_7457_arctan__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_7458_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_7459_set__vebt__pred_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( vEBT_vebt_pred(T,Xa) = $ite(
            ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),vEBT_set_vebt(T))
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),Xa) ),
            some_nat(order_Greatest_nat(aa_nat_fun_nat_o(aTP_Lamm_wp(T),Xa))),
            none_nat ) ) ) ).

% set_vebt_pred'
tff(fact_7460_arctan__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% arctan_less_iff
tff(fact_7461_arctan__monotone,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone
tff(fact_7462_arctan__monotone_H,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone'
tff(fact_7463_arctan__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xa)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% arctan_le_iff
tff(fact_7464_finite__int__iff__bounded__le,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(abs_abs_int),S2)),set_ord_atMost_int(K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_7465_finite__int__iff__bounded,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(abs_abs_int),S2)),set_ord_lessThan_int(K3)) ) ).

% finite_int_iff_bounded
tff(fact_7466_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ! [M5: int] :
        ? [N5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M5),aa_int_int(abs_abs_int,N5))
          & aa_set_int_o(member_int(N5),S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_7467_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ! [M5: int] :
        ? [N5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M5),aa_int_int(abs_abs_int,N5))
          & aa_set_int_o(member_int(N5),S2) ) ) ).

% infinite_int_iff_unbounded
tff(fact_7468_zabs__def,axiom,
    ! [I: int] :
      ( aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ) ).

% zabs_def
tff(fact_7469_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_7470_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7471_incr__lemma,axiom,
    ! [D: int,Z: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),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,Xa),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_7472_decr__lemma,axiom,
    ! [D: int,Xa: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),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,Xa),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_7473_nat__ivt__aux,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7474_sin__coeff__Suc,axiom,
    ! [Na: nat] : ( aa_nat_real(sin_coeff,aa_nat_nat(suc,Na)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ) ).

% sin_coeff_Suc
tff(fact_7475_nat0__intermed__int__val,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7476_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ) ).

% VEBT.size(3)
tff(fact_7477_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_size_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ) ).

% VEBT.size_gen(1)
tff(fact_7478_ln__inj__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( ( aa_real_real(ln_ln_real,Xa) = aa_real_real(ln_ln_real,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_7479_ln__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_7480_ln__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_7481_ln__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( ( aa_real_real(ln_ln_real,Xa) = zero_zero_real )
      <=> ( Xa = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_7482_ln__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% ln_gt_zero_iff
tff(fact_7483_ln__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_7484_exp__ln,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xa)) = Xa ) ) ).

% exp_ln
tff(fact_7485_exp__ln__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xa)) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% exp_ln_iff
tff(fact_7486_ln__ge__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% ln_ge_zero_iff
tff(fact_7487_ln__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_7488_ln__less__self,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),Xa) ) ).

% ln_less_self
tff(fact_7489_GreatestI__ex__nat,axiom,
    ! [P2: fun_nat_o,B: nat] :
      ( ? [X_13: nat] : aa_nat_o(P2,X_13)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P2,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(P2,order_Greatest_nat(P2)) ) ) ).

% GreatestI_ex_nat
tff(fact_7490_Greatest__le__nat,axiom,
    ! [P2: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P2,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P2,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P2)) ) ) ).

% Greatest_le_nat
tff(fact_7491_GreatestI__nat,axiom,
    ! [P2: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P2,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P2,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B) )
       => aa_nat_o(P2,order_Greatest_nat(P2)) ) ) ).

% GreatestI_nat
tff(fact_7492_ln__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),Xa) ) ).

% ln_bound
tff(fact_7493_ln__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa)) ) ).

% ln_gt_zero
tff(fact_7494_ln__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xa)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_7495_ln__gt__zero__imp__gt__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_7496_ln__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa)) ) ).

% ln_ge_zero
tff(fact_7497_ln__ge__zero__imp__ge__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_7498_ln__add__one__self__le__self,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa) ) ).

% ln_add_one_self_le_self
tff(fact_7499_ln__mult,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_7500_ln__eq__minus__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( ( aa_real_real(ln_ln_real,Xa) = aa_real_real(minus_minus_real(Xa),one_one_real) )
       => ( Xa = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_7501_ln__div,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(Xa),Y)) = aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_7502_ln__ge__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,Xa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xa) ) ) ).

% ln_ge_iff
tff(fact_7503_ln__x__over__x__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Xa)),Xa)) ) ) ).

% ln_x_over_x_mono
tff(fact_7504_ln__le__minus__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(minus_minus_real(Xa),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_7505_ln__diff__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,Xa)),aa_real_real(ln_ln_real,Y))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xa),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_7506_ln__add__one__self__le__self2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa) ) ).

% ln_add_one_self_le_self2
tff(fact_7507_ln__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_realpow
tff(fact_7508_ln__one__minus__pos__upper__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),Xa))),aa_real_real(uminus_uminus_real,Xa)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_7509_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P2: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(A0,A12))
     => ( ! [I2: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I2,J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P2,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P2,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P2,A0),A12) ) ) ).

% upto.pinduct
tff(fact_7510_card__length__sum__list__rec,axiom,
    ! [M: nat,N6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_wq(M),N6))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_wr(M),N6)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ws(M),N6)))) ) ) ).

% card_length_sum_list_rec
tff(fact_7511_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q3,R))
    <=> ( ( 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)),R) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int) ),
              Q3 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_7512_bezw__0,axiom,
    ! [Xa: nat] : ( bezw(Xa,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ) ).

% bezw_0
tff(fact_7513_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R: int,Q6: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),product_Pair_int_int(Q3,R))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),product_Pair_int_int(Q6,R4))
       => ( Q3 = Q6 ) ) ) ).

% unique_quotient
tff(fact_7514_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R: int,Q6: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),product_Pair_int_int(Q3,R))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),product_Pair_int_int(Q6,R4))
       => ( R = R4 ) ) ) ).

% unique_remainder
tff(fact_7515_eucl__rel__int__by0,axiom,
    ! [K: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,zero_zero_int),product_Pair_int_int(zero_zero_int,K)) ).

% eucl_rel_int_by0
tff(fact_7516_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q3,R))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = R ) ) ).

% mod_int_unique
tff(fact_7517_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q3,R))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q3 ) ) ).

% div_int_unique
tff(fact_7518_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q3),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q3,zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_7519_eucl__rel__int,axiom,
    ! [K: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ).

% eucl_rel_int
tff(fact_7520_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),product_Pair_int_int(Q3,R))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            product_Pair_int_int($ite(R = zero_zero_int,aa_int_int(uminus_uminus_int,Q3),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q3)),one_one_int)),
              $ite(R = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R)))) ) ) ).

% zminus1_lemma
tff(fact_7521_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [R3: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S2)))
          & ! [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N7),aa_set_nat_nat(finite_card_nat,S2))
             => aa_set_nat_o(member_nat(aa_nat_nat(R3,N7)),S2) ) ) ) ).

% finite_enumerate
tff(fact_7522_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bs,aTP_Lamm_br) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_7523_Suc__funpow,axiom,
    ! [Na: nat] : ( compow_fun_nat_nat(Na,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Na) ) ).

% Suc_funpow
tff(fact_7524_eucl__rel__int_Osimps,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A12,A23),A32)
    <=> ( ? [K3: int] :
            ( ( A12 = K3 )
            & ( A23 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L4: int,K3: int,Q7: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = product_Pair_int_int(Q7,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q7),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q7: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = product_Pair_int_int(Q7,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q7),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_7525_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A12,A23),A32)
     => ( ( ( A23 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A12) ) )
       => ( ! [Q5: int] :
              ( ( A32 = product_Pair_int_int(Q5,zero_zero_int) )
             => ( ( A23 != zero_zero_int )
               => ( A12 != aa_int_int(aa_int_fun_int_int(times_times_int,Q5),A23) ) ) )
         => ~ ! [R3: int,Q5: int] :
                ( ( A32 = product_Pair_int_int(Q5,R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A23) )
                 => ( aa_int_o(aa_int_fun_int_o(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,Q5),A23)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_7526_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_7527_zsgn__def,axiom,
    ! [I: int] :
      ( aa_int_int(sgn_sgn_int,I) = $ite(
          I = zero_zero_int,
          zero_zero_int,
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ) ).

% zsgn_def
tff(fact_7528_div__sgn__abs__cancel,axiom,
    ! [V2: int,K: int,L: int] :
      ( ( V2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V2)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V2)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_7529_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q3: int] :
      ( ( aa_int_int(sgn_sgn_int,R) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R)),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)),R) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q3,R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_7530_sinh__real__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% sinh_real_less_iff
tff(fact_7531_sinh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% sinh_real_le_iff
tff(fact_7532_sinh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% sinh_real_pos_iff
tff(fact_7533_sinh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_7534_sgn__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_7535_zero__le__sgn__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% zero_le_sgn_iff
tff(fact_7536_sinh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_7537_sinh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% sinh_real_nonneg_iff
tff(fact_7538_sgn__real__def,axiom,
    ! [A: real] :
      ( aa_real_real(sgn_sgn_real,A) = $ite(
          A = zero_zero_real,
          zero_zero_real,
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ) ).

% sgn_real_def
tff(fact_7539_sgn__power__injE,axiom,
    ! [A: real,Na: nat,Xa: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Na)) = Xa )
     => ( ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Na)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_7540_divmod__nat__def,axiom,
    ! [M: nat,Na: nat] : ( divmod_nat(M,Na) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) ) ).

% divmod_nat_def
tff(fact_7541_ln__root,axiom,
    ! [Na: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Na),B)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% ln_root
tff(fact_7542_real__root__Suc__0,axiom,
    ! [Xa: real] : ( aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xa) = Xa ) ).

% real_root_Suc_0
tff(fact_7543_real__root__eq__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = aa_real_real(aa_nat_fun_real_real(root,Na),Y) )
      <=> ( Xa = Y ) ) ) ).

% real_root_eq_iff
tff(fact_7544_root__0,axiom,
    ! [Xa: real] : ( aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xa) = zero_zero_real ) ).

% root_0
tff(fact_7545_real__root__eq__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = zero_zero_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_7546_real__root__less__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ).

% real_root_less_iff
tff(fact_7547_real__root__le__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ).

% real_root_le_iff
tff(fact_7548_real__root__eq__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = one_one_real )
      <=> ( Xa = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_7549_real__root__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_7550_real__root__lt__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_7551_real__root__gt__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_7552_real__root__le__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_7553_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_7554_real__root__lt__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_7555_real__root__gt__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_7556_real__root__le__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_7557_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_7558_real__root__pow__pos2,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos2
tff(fact_7559_sgn__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) = aa_real_real(sgn_sgn_real,Xa) ) ) ).

% sgn_root
tff(fact_7560_real__root__pos__pos__le,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ).

% real_root_pos_pos_le
tff(fact_7561_real__root__less__mono,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_less_mono
tff(fact_7562_real__root__le__mono,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_7563_real__root__power,axiom,
    ! [Na: nat,Xa: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),K) ) ) ).

% real_root_power
tff(fact_7564_real__root__abs,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(abs_abs_real,Xa)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ).

% real_root_abs
tff(fact_7565_real__root__gt__zero,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ).

% real_root_gt_zero
tff(fact_7566_real__root__strict__decreasing,axiom,
    ! [Na: nat,N6: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N6),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_7567_root__abs__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_7568_real__root__pos__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ).

% real_root_pos_pos
tff(fact_7569_real__root__strict__increasing,axiom,
    ! [Na: nat,N6: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,N6),Xa)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_7570_real__root__decreasing,axiom,
    ! [Na: nat,N6: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N6),Xa)),aa_real_real(aa_nat_fun_real_real(root,Na),Xa)) ) ) ) ).

% real_root_decreasing
tff(fact_7571_real__root__pow__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos
tff(fact_7572_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na) = Xa )
         => ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_7573_real__root__power__cancel,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = Xa ) ) ) ).

% real_root_power_cancel
tff(fact_7574_sgn__power__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa))),Na)) = Xa ) ) ).

% sgn_power_root
tff(fact_7575_root__sgn__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),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)),Na))) = Y ) ) ).

% root_sgn_power
tff(fact_7576_real__root__increasing,axiom,
    ! [Na: nat,N6: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_real_real(aa_nat_fun_real_real(root,N6),Xa)) ) ) ) ) ).

% real_root_increasing
tff(fact_7577_split__root,axiom,
    ! [P2: fun_real_o,Na: nat,Xa: real] :
      ( aa_real_o(P2,aa_real_real(aa_nat_fun_real_real(root,Na),Xa))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_real_o(P2,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ! [Y5: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y5)),Na)) = Xa )
             => aa_real_o(P2,Y5) ) ) ) ) ).

% split_root
tff(fact_7578_root__powr__inverse,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = aa_real_real(powr_real(Xa),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ) ).

% root_powr_inverse
tff(fact_7579_log__base__root,axiom,
    ! [Na: nat,B: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(log(aa_real_real(aa_nat_fun_real_real(root,Na),B)),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(B),Xa)) ) ) ) ).

% log_base_root
tff(fact_7580_log__root,axiom,
    ! [Na: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(log(B),aa_real_real(aa_nat_fun_real_real(root,Na),A)) = aa_real_real(divide_divide_real(aa_real_real(log(B),A)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_root
tff(fact_7581_powr__gt__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(powr_real(Xa),A))
    <=> ( Xa != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_7582_powr__nonneg__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(A),Xa)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_7583_powr__less__cancel__iff,axiom,
    ! [Xa: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel_iff
tff(fact_7584_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(log(A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_7585_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),Xa)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_7586_one__less__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(log(A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xa) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_7587_log__less__one__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),Xa)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_7588_log__less__cancel__iff,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),Xa)),aa_real_real(log(A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_7589_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_7590_powr__eq__one__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(powr_real(A),Xa) = one_one_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_7591_powr__one__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( ( aa_real_real(powr_real(Xa),one_one_real) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% powr_one_gt_zero_iff
tff(fact_7592_powr__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),one_one_real) = Xa ) ) ).

% powr_one
tff(fact_7593_powr__le__cancel__iff,axiom,
    ! [Xa: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_7594_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(log(A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_7595_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),Xa)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_7596_one__le__log__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(log(A),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xa) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_7597_log__le__one__cancel__iff,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),Xa)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_7598_log__le__cancel__iff,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),Xa)),aa_real_real(log(A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_7599_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_real_real(powr_real(A),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_7600_powr__log__cancel,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(powr_real(A),aa_real_real(log(A),Xa)) = Xa ) ) ) ) ).

% powr_log_cancel
tff(fact_7601_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_7602_less__log__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(log(B),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),Y)),Xa) ) ) ) ).

% less_log_iff
tff(fact_7603_log__less__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),Xa)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(powr_real(B),Y)) ) ) ) ).

% log_less_iff
tff(fact_7604_less__powr__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(powr_real(B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),Xa)),Y) ) ) ) ).

% less_powr_iff
tff(fact_7605_powr__less__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),Y)),Xa)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(log(B),Xa)) ) ) ) ).

% powr_less_iff
tff(fact_7606_le__log__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(log(B),Xa))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),Y)),Xa) ) ) ) ).

% le_log_iff
tff(fact_7607_log__le__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),Xa)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(powr_real(B),Y)) ) ) ) ).

% log_le_iff
tff(fact_7608_le__powr__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(powr_real(B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),Xa)),Y) ) ) ) ).

% le_powr_iff
tff(fact_7609_powr__le__iff,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),Y)),Xa)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(log(B),Xa)) ) ) ) ).

% powr_le_iff
tff(fact_7610_powr__less__mono2__neg,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Y),A)),aa_real_real(powr_real(Xa),A)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_7611_powr__non__neg,axiom,
    ! [A: real,Xa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(A),Xa)),zero_zero_real) ).

% powr_non_neg
tff(fact_7612_powr__mono2,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_mono2
tff(fact_7613_powr__ge__pzero,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(powr_real(Xa),Y)) ).

% powr_ge_pzero
tff(fact_7614_powr__less__cancel,axiom,
    ! [Xa: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel
tff(fact_7615_powr__less__mono,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B)) ) ) ).

% powr_less_mono
tff(fact_7616_powr__mono,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Xa),B)) ) ) ).

% powr_mono
tff(fact_7617_add__log__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(log(B),Xa)) = aa_real_real(log(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(B),Y)),Xa)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_7618_log__add__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(B),Xa)),Y) = aa_real_real(log(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(powr_real(B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_7619_minus__log__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(minus_minus_real(Y),aa_real_real(log(B),Xa)) = aa_real_real(log(B),aa_real_real(divide_divide_real(aa_real_real(powr_real(B),Y)),Xa)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_7620_log__minus__eq__powr,axiom,
    ! [B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(minus_minus_real(aa_real_real(log(B),Xa)),Y) = aa_real_real(log(B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_7621_powr__less__mono2,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_less_mono2
tff(fact_7622_powr__mono2_H,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Y),A)),aa_real_real(powr_real(Xa),A)) ) ) ) ).

% powr_mono2'
tff(fact_7623_gr__one__powr,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(powr_real(Xa),Y)) ) ) ).

% gr_one_powr
tff(fact_7624_powr__inj,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(powr_real(A),Xa) = aa_real_real(powr_real(A),Y) )
        <=> ( Xa = Y ) ) ) ) ).

% powr_inj
tff(fact_7625_ge__one__powr__ge__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(powr_real(Xa),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_7626_powr__mono__both,axiom,
    ! [A: real,B: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_7627_powr__le1,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xa),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_7628_powr__divide,axiom,
    ! [Xa: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(powr_real(aa_real_real(divide_divide_real(Xa),Y)),A) = aa_real_real(divide_divide_real(aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_divide
tff(fact_7629_powr__mult,axiom,
    ! [Xa: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(Xa),A)),aa_real_real(powr_real(Y),A)) ) ) ) ).

% powr_mult
tff(fact_7630_log__base__change,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(B),Xa) = aa_real_real(divide_divide_real(aa_real_real(log(A),Xa)),aa_real_real(log(A),B)) ) ) ) ).

% log_base_change
tff(fact_7631_powr__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na) ) ) ).

% powr_realpow
tff(fact_7632_log__of__power__eq,axiom,
    ! [M: nat,B: real,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_7633_less__log__of__power,axiom,
    ! [B: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(B),M)) ) ) ).

% less_log_of_power
tff(fact_7634_powr__neg__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(divide_divide_real(one_one_real),Xa) ) ) ).

% powr_neg_one
tff(fact_7635_powr__mult__base,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),aa_real_real(powr_real(Xa),Y)) = aa_real_real(powr_real(Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_7636_log__mult,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(log(A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(A),Xa)),aa_real_real(log(A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_7637_le__log__of__power,axiom,
    ! [B: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(B),M)) ) ) ).

% le_log_of_power
tff(fact_7638_log__base__pow,axiom,
    ! [A: real,Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(log(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Na)),Xa) = aa_real_real(divide_divide_real(aa_real_real(log(A),Xa)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log_base_pow
tff(fact_7639_log__nat__power,axiom,
    ! [Xa: real,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(log(B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(B),Xa)) ) ) ).

% log_nat_power
tff(fact_7640_log__divide,axiom,
    ! [A: real,Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(log(A),aa_real_real(divide_divide_real(Xa),Y)) = aa_real_real(minus_minus_real(aa_real_real(log(A),Xa)),aa_real_real(log(A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_7641_log__of__power__less,axiom,
    ! [M: nat,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_less
tff(fact_7642_ln__powr__bound,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xa)),aa_real_real(divide_divide_real(aa_real_real(powr_real(Xa),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_7643_ln__powr__bound2,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(aa_real_real(ln_ln_real,Xa)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(A),A)),Xa)) ) ) ).

% ln_powr_bound2
tff(fact_7644_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
             => ( aa_real_real(log(A),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(log(B),Xa)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_7645_log__of__power__le,axiom,
    ! [M: nat,B: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_le
tff(fact_7646_ceiling__log__eq__powr__iff,axiom,
    ! [Xa: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(B),Xa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_7647_real__of__int__div4,axiom,
    ! [Na: int,Xa: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xa))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,Xa))) ).

% real_of_int_div4
tff(fact_7648_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),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_7649_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real)),aa_int_real(ring_1_of_int_real,M)) ) ).

% int_less_real_le
tff(fact_7650_real__of__int__div2,axiom,
    ! [Na: int,Xa: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,Xa))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xa)))) ).

% real_of_int_div2
tff(fact_7651_real__of__int__div3,axiom,
    ! [Na: int,Xa: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,Xa))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xa)))),one_one_real) ).

% real_of_int_div3
tff(fact_7652_powr__int,axiom,
    ! [Xa: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,I)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_7653_floor__log__eq__powr__iff,axiom,
    ! [Xa: real,B: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(B),Xa)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),aa_int_real(ring_1_of_int_real,K))),Xa)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(powr_real(B),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_7654_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_7655_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_7656_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_7657_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% zless_nat_conj
tff(fact_7658_nat__zminus__int,axiom,
    ! [Na: nat] : ( aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))) = zero_zero_nat ) ).

% nat_zminus_int
tff(fact_7659_int__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ) ).

% int_nat_eq
tff(fact_7660_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_7661_nat__ceiling__le__eq,axiom,
    ! [Xa: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xa))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_7662_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_7663_nat__floor__neg,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_7664_floor__eq3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = Na ) ) ) ).

% floor_eq3
tff(fact_7665_le__nat__floor,axiom,
    ! [Xa: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xa)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_7666_floor__eq4,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xa)) = Na ) ) ) ).

% floor_eq4
tff(fact_7667_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_7668_nat__mono,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_7669_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z6) )
        <=> ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_7670_all__nat,axiom,
    ! [P2: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(P2,X_12)
    <=> ! [X5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => aa_nat_o(P2,aa_int_nat(nat2,X5)) ) ) ).

% all_nat
tff(fact_7671_ex__nat,axiom,
    ! [P2: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P2,X_12)
    <=> ? [X5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
          & aa_nat_o(P2,aa_int_nat(nat2,X5)) ) ) ).

% ex_nat
tff(fact_7672_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_7673_floor__eq,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xa) = Na ) ) ) ).

% floor_eq
tff(fact_7674_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_7675_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))) ).

% real_of_int_floor_gt_diff_one
tff(fact_7676_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))) ).

% real_of_int_floor_ge_diff_one
tff(fact_7677_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_mono_iff
tff(fact_7678_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_7679_nat__le__iff,axiom,
    ! [Xa: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xa)),Na)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_le_iff
tff(fact_7680_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_7681_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_7682_real__nat__ceiling__ge,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xa)))) ).

% real_nat_ceiling_ge
tff(fact_7683_floor__eq2,axiom,
    ! [Na: int,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Na)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xa) = Na ) ) ) ).

% floor_eq2
tff(fact_7684_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_7685_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_less_eq_zless
tff(fact_7686_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa_int_nat(nat2,W2) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_7687_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_7688_split__nat,axiom,
    ! [P2: fun_nat_o,I: int] :
      ( aa_nat_o(P2,aa_int_nat(nat2,I))
    <=> ( ! [N5: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N5) )
           => aa_nat_o(P2,N5) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P2,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_7689_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W2)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ) ).

% nat_le_eq_zle
tff(fact_7690_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
       => ( aa_int_nat(nat2,aa_int_int(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_7691_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ) ).

% le_nat_iff
tff(fact_7692_Suc__as__int,axiom,
    ! [X3: nat] : ( aa_nat_nat(suc,X3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),one_one_int)) ) ).

% Suc_as_int
tff(fact_7693_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(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_7694_nat__diff__distrib_H,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_7695_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z6),Z)
       => ( aa_int_nat(nat2,aa_int_int(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_7696_nat__div__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_7697_nat__div__distrib_H,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_7698_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_7699_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_7700_nat__mod__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xa)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_7701_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) ).

% div_abs_eq_div_nat
tff(fact_7702_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ) ).

% mod_abs_eq_div_nat
tff(fact_7703_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_7704_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_7705_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(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_7706_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ) ).

% nat_abs_int_diff
tff(fact_7707_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = aa_set_nat_set_int(image_nat_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_7708_diff__nat__eq__if,axiom,
    ! [Z: int,Z6: int] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,Z6),zero_zero_int),
          aa_int_nat(nat2,Z),
          $let(
            d: int,
            d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6),
            $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ) ).

% diff_nat_eq_if
tff(fact_7709_powr__real__of__int,axiom,
    ! [Xa: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_int_real(ring_1_of_int_real,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,Na)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Na))))) ) ) ).

% powr_real_of_int
tff(fact_7710_floor__real__def,axiom,
    ! [Xa: real] : ( aa_real_int(archim6058952711729229775r_real,Xa) = the_int(aa_real_fun_int_o(aTP_Lamm_wt,Xa)) ) ).

% floor_real_def
tff(fact_7711_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_real(powr_real(aa_real_real(inverse_inverse_real,Y)),A) = aa_real_real(inverse_inverse_real,aa_real_real(powr_real(Y),A)) ) ) ).

% inverse_powr
tff(fact_7712_forall__pos__mono__1,axiom,
    ! [P2: fun_real_o,E2: real] :
      ( ! [D2: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E)
         => ( aa_real_o(P2,D2)
           => aa_real_o(P2,E) ) )
     => ( ! [N: nat] : aa_real_o(P2,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P2,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_7713_forall__pos__mono,axiom,
    ! [P2: fun_real_o,E2: real] :
      ( ! [D2: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E)
         => ( aa_real_o(P2,D2)
           => aa_real_o(P2,E) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => aa_real_o(P2,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P2,E2) ) ) ) ).

% forall_pos_mono
tff(fact_7714_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
    <=> ? [N5: nat] :
          ( ( N5 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N5)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N5))),E2) ) ) ).

% real_arch_inverse
tff(fact_7715_ln__inverse,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xa)) ) ) ).

% ln_inverse
tff(fact_7716_log__inverse,axiom,
    ! [A: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( aa_real_real(log(A),aa_real_real(inverse_inverse_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(log(A),Xa)) ) ) ) ) ).

% log_inverse
tff(fact_7717_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_7718_Inf__nat__def1,axiom,
    ! [K4: set_nat] :
      ( ( K4 != bot_bot_set_nat )
     => aa_set_nat_o(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K4)),K4) ) ).

% Inf_nat_def1
tff(fact_7719_Sup__nat__def,axiom,
    ! [X4: set_nat] :
      ( aa_set_nat_nat(complete_Sup_Sup_nat,X4) = $ite(X4 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X4)) ) ).

% Sup_nat_def
tff(fact_7720_ln__neg__is__const,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xa) = the_real(aTP_Lamm_wu) ) ) ).

% ln_neg_is_const
tff(fact_7721_floor__rat__def,axiom,
    ! [Xa: rat] : ( aa_rat_int(archim3151403230148437115or_rat,Xa) = the_int(aTP_Lamm_wv(Xa)) ) ).

% floor_rat_def
tff(fact_7722_less__eq__rat__def,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_rat_def
tff(fact_7723_sgn__rat__def,axiom,
    ! [A: rat] :
      ( aa_rat_rat(sgn_sgn_rat,A) = $ite(
          A = zero_zero_rat,
          zero_zero_rat,
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ) ).

% sgn_rat_def
tff(fact_7724_abs__rat__def,axiom,
    ! [A: rat] :
      ( aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ) ).

% abs_rat_def
tff(fact_7725_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R)
     => ~ ! [S3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S3)
           => ! [T6: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T6)
               => ( R != aa_rat_rat(plus_plus_rat(S3),T6) ) ) ) ) ).

% obtain_pos_sum
tff(fact_7726_Maclaurin__sin__bound,axiom,
    ! [Xa: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(sin_real,Xa)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ww,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xa)),Na))) ).

% Maclaurin_sin_bound
tff(fact_7727_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(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_7728_Cauchy__iff2,axiom,
    ! [X4: fun_nat_real] :
      ( topolo4055970368930404560y_real(X4)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M5)
         => ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(X4,M5)),aa_nat_real(X4,N5)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_7729_sin__x__le__x,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),Xa) ) ).

% sin_x_le_x
tff(fact_7730_sin__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),one_one_real) ).

% sin_le_one
tff(fact_7731_abs__sin__x__le__abs__x,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xa))),aa_real_real(abs_abs_real,Xa)) ).

% abs_sin_x_le_abs_x
tff(fact_7732_sin__x__ge__neg__x,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xa)),aa_real_real(sin_real,Xa)) ) ).

% sin_x_ge_neg_x
tff(fact_7733_sin__ge__minus__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,Xa)) ).

% sin_ge_minus_one
tff(fact_7734_abs__sin__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xa))),one_one_real) ).

% abs_sin_le_one
tff(fact_7735_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P3: int,Q3: int] :
      ( ( normalize(R) = product_Pair_int_int(P3,Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% normalize_denom_pos
tff(fact_7736_mod__or__dist,axiom,
    ! [P2: assn,Q2: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(sup_sup_assn(P2,Q2)),H2)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H2)
        | aa_Pro653082453696054616_nat_o(rep_assn(Q2),H2) ) ) ).

% mod_or_dist
tff(fact_7737_merge__pure__or,axiom,
    ! [A: $o,B: $o] :
      ( sup_sup_assn(aa_o_assn(pure_assn,(A)),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
          ( (A)
          | (B) )) ) ).

% merge_pure_or
tff(fact_7738_mod__h__bot__iff_I7_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(sup_sup_assn(P2,Q2)),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P2),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
        | aa_Pro653082453696054616_nat_o(rep_assn(Q2),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(7)
tff(fact_7739_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_7740_ent__disjE,axiom,
    ! [A4: assn,C3: assn,B4: assn] :
      ( entails(A4,C3)
     => ( entails(B4,C3)
       => entails(sup_sup_assn(A4,B4),C3) ) ) ).

% ent_disjE
tff(fact_7741_ent__disjI1,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( entails(sup_sup_assn(P2,Q2),R2)
     => entails(P2,R2) ) ).

% ent_disjI1
tff(fact_7742_ent__disjI2,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( entails(sup_sup_assn(P2,Q2),R2)
     => entails(Q2,R2) ) ).

% ent_disjI2
tff(fact_7743_ent__disjI1_H,axiom,
    ! [A4: assn,B4: assn,C3: assn] :
      ( entails(A4,B4)
     => entails(A4,sup_sup_assn(B4,C3)) ) ).

% ent_disjI1'
tff(fact_7744_ent__disjI2_H,axiom,
    ! [A4: assn,C3: assn,B4: assn] :
      ( entails(A4,C3)
     => entails(A4,sup_sup_assn(B4,C3)) ) ).

% ent_disjI2'
tff(fact_7745_ent__disjI1__direct,axiom,
    ! [A4: assn,B4: assn] : entails(A4,sup_sup_assn(A4,B4)) ).

% ent_disjI1_direct
tff(fact_7746_ent__disjI2__direct,axiom,
    ! [B4: assn,A4: assn] : entails(B4,sup_sup_assn(A4,B4)) ).

% ent_disjI2_direct
tff(fact_7747_star__or__dist1,axiom,
    ! [A4: assn,B4: assn,C3: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sup_sup_assn(A4,B4)),C3) = sup_sup_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),C3),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B4),C3)) ) ).

% star_or_dist1
tff(fact_7748_star__or__dist2,axiom,
    ! [C3: assn,A4: assn,B4: assn] : ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),sup_sup_assn(A4,B4)) = sup_sup_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),A4),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C3),B4)) ) ).

% star_or_dist2
tff(fact_7749_infinite__UNIV__nat,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% infinite_UNIV_nat
tff(fact_7750_nat__not__finite,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% nat_not_finite
tff(fact_7751_mod__star__conv,axiom,
    ! [A4: assn,B4: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),B4)),H2)
    <=> ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As2: set_nat] :
          ( ( H2 = produc7507926704131184380et_nat(Hr,sup_sup_set_nat(As1,As2)) )
          & ( inf_inf_set_nat(As1,As2) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(rep_assn(A4),produc7507926704131184380et_nat(Hr,As1))
          & aa_Pro653082453696054616_nat_o(rep_assn(B4),produc7507926704131184380et_nat(Hr,As2)) ) ) ).

% mod_star_conv
tff(fact_7752_star__assnI,axiom,
    ! [P2: assn,H2: heap_e7401611519738050253t_unit,As3: set_nat,Q2: assn,As4: set_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P2),produc7507926704131184380et_nat(H2,As3))
     => ( aa_Pro653082453696054616_nat_o(rep_assn(Q2),produc7507926704131184380et_nat(H2,As4))
       => ( ( inf_inf_set_nat(As3,As4) = bot_bot_set_nat )
         => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Q2)),produc7507926704131184380et_nat(H2,sup_sup_set_nat(As3,As4))) ) ) ) ).

% star_assnI
tff(fact_7753_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_7754_range__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_set_nat(image_nat_nat(infini8530281810654367211te_nat(S2)),top_top_set_nat) = S2 ) ) ).

% range_enumerate
tff(fact_7755_UNIV__nat__eq,axiom,
    top_top_set_nat = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),top_top_set_nat)) ).

% UNIV_nat_eq
tff(fact_7756_range__mod,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_wx(Na)),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Na) ) ) ).

% range_mod
tff(fact_7757_card__nat,axiom,
    aa_set_nat_nat(finite_card_nat,top_top_set_nat) = zero_zero_nat ).

% card_nat
tff(fact_7758_card__num0,axiom,
    finite6454714172617411596l_num0(top_to3689904424835650196l_num0) = zero_zero_nat ).

% card_num0
tff(fact_7759_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_7760_merge__true__star,axiom,
    aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_7761_merge__pure__and,axiom,
    ! [A: $o,B: $o] :
      ( inf_inf_assn(aa_o_assn(pure_assn,(A)),aa_o_assn(pure_assn,(B))) = aa_o_assn(pure_assn,
          ( (A)
          & (B) )) ) ).

% merge_pure_and
tff(fact_7762_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_7763_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_7764_mod__h__bot__iff_I6_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(inf_inf_assn(P2,Q2)),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P2),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(rep_assn(Q2),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(6)
tff(fact_7765_card__literal,axiom,
    finite_card_literal(top_top_set_literal) = zero_zero_nat ).

% card_literal
tff(fact_7766_ent__conjE2,axiom,
    ! [B4: assn,C3: assn,A4: assn] :
      ( entails(B4,C3)
     => entails(inf_inf_assn(A4,B4),C3) ) ).

% ent_conjE2
tff(fact_7767_ent__conjE1,axiom,
    ! [A4: assn,C3: assn,B4: assn] :
      ( entails(A4,C3)
     => entails(inf_inf_assn(A4,B4),C3) ) ).

% ent_conjE1
tff(fact_7768_ent__conjI,axiom,
    ! [A4: assn,B4: assn,C3: assn] :
      ( entails(A4,B4)
     => ( entails(A4,C3)
       => entails(A4,inf_inf_assn(B4,C3)) ) ) ).

% ent_conjI
tff(fact_7769_mod__and__dist,axiom,
    ! [P2: assn,Q2: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(inf_inf_assn(P2,Q2)),H2)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H2)
        & aa_Pro653082453696054616_nat_o(rep_assn(Q2),H2) ) ) ).

% mod_and_dist
tff(fact_7770_ent__true,axiom,
    ! [P2: assn] : entails(P2,top_top_assn) ).

% ent_true
tff(fact_7771_mod__star__trueE,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),top_top_assn)),H2)
     => ~ ! [H5: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(P2),H5) ) ).

% mod_star_trueE
tff(fact_7772_mod__star__trueI,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P2),H2)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),top_top_assn)),H2) ) ).

% mod_star_trueI
tff(fact_7773_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(top_top_assn),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).

% mod_h_bot_iff(2)
tff(fact_7774_root__def,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_real_real(aa_nat_fun_real_real(root,Na),Xa) = $ite(Na = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_wy,Na)),Xa)) ) ).

% root_def
tff(fact_7775_less__eq__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_eq_assn(A,B)
    <=> ( A = inf_inf_assn(A,B) ) ) ).

% less_eq_assn_def
tff(fact_7776_minus__assn__def,axiom,
    ! [A: assn,B: assn] : ( minus_minus_assn(A,B) = inf_inf_assn(A,uminus_uminus_assn(B)) ) ).

% minus_assn_def
tff(fact_7777_times__assn__raw_Oelims_I3_J,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Xaa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(Xa),Xaa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ? [As12: set_nat,As22: set_nat] :
                ( ( As = sup_sup_set_nat(As12,As22) )
                & ( inf_inf_set_nat(As12,As22) = bot_bot_set_nat )
                & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As12))
                & aa_Pro653082453696054616_nat_o(Xaa,produc7507926704131184380et_nat(H,As22)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_7778_less__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( ord_less_assn(A,B)
    <=> ( ord_less_eq_assn(A,B)
        & ( A != B ) ) ) ).

% less_assn_def
tff(fact_7779_binomial__def,axiom,
    ! [Na: nat,K: nat] : ( aa_nat_nat(binomial(Na),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_wz(Na),K))) ) ).

% binomial_def
tff(fact_7780_times__assn__raw_Osimps,axiom,
    ! [P2: fun_Pr6460944699146269935_nat_o,Q2: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As3: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(P2),Q2),produc7507926704131184380et_nat(H2,As3))
    <=> ? [As1: set_nat,As2: set_nat] :
          ( ( As3 = sup_sup_set_nat(As1,As2) )
          & ( inf_inf_set_nat(As1,As2) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(P2,produc7507926704131184380et_nat(H2,As1))
          & aa_Pro653082453696054616_nat_o(Q2,produc7507926704131184380et_nat(H2,As2)) ) ) ).

% times_assn_raw.simps
tff(fact_7781_times__assn__raw_Oelims_I1_J,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Xaa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(Xa),Xaa),Xb)
      <=> (Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ( (Y)
            <=> ~ ? [As1: set_nat,As2: set_nat] :
                    ( ( As = sup_sup_set_nat(As1,As2) )
                    & ( inf_inf_set_nat(As1,As2) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As1))
                    & aa_Pro653082453696054616_nat_o(Xaa,produc7507926704131184380et_nat(H,As2)) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_7782_times__assn__raw_Oelims_I2_J,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Xaa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(Xa),Xaa),Xb)
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ~ ? [As13: set_nat,As23: set_nat] :
                  ( ( As = sup_sup_set_nat(As13,As23) )
                  & ( inf_inf_set_nat(As13,As23) = bot_bot_set_nat )
                  & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As13))
                  & aa_Pro653082453696054616_nat_o(Xaa,produc7507926704131184380et_nat(H,As23)) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_7783_times__assn__raw_Opelims_I1_J,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Xaa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(Xa),Xaa),Xb)
      <=> (Y) )
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(Xa,produc6017135503889078682et_nat(Xaa,Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( ( (Y)
                <=> ? [As1: set_nat,As2: set_nat] :
                      ( ( As = sup_sup_set_nat(As1,As2) )
                      & ( inf_inf_set_nat(As1,As2) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As1))
                      & aa_Pro653082453696054616_nat_o(Xaa,produc7507926704131184380et_nat(H,As2)) ) )
               => ~ aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(Xa,produc6017135503889078682et_nat(Xaa,produc7507926704131184380et_nat(H,As)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_7784_times__assn__raw_Opelims_I2_J,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Xaa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(Xa),Xaa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(Xa,produc6017135503889078682et_nat(Xaa,Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(Xa,produc6017135503889078682et_nat(Xaa,produc7507926704131184380et_nat(H,As))))
               => ~ ? [As13: set_nat,As23: set_nat] :
                      ( ( As = sup_sup_set_nat(As13,As23) )
                      & ( inf_inf_set_nat(As13,As23) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As13))
                      & aa_Pro653082453696054616_nat_o(Xaa,produc7507926704131184380et_nat(H,As23)) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_7785_times__assn__raw_Opelims_I3_J,axiom,
    ! [Xa: fun_Pr6460944699146269935_nat_o,Xaa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(times_assn_raw(Xa),Xaa),Xb)
     => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(Xa,produc6017135503889078682et_nat(Xaa,Xb)))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( aa_Pro2671238023309943196_nat_o(accp_P1394777952347926645et_nat(times_assn_raw_rel),produc7291855603632349880et_nat(Xa,produc6017135503889078682et_nat(Xaa,produc7507926704131184380et_nat(H,As))))
               => ? [As12: set_nat,As22: set_nat] :
                    ( ( As = sup_sup_set_nat(As12,As22) )
                    & ( inf_inf_set_nat(As12,As22) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H,As12))
                    & aa_Pro653082453696054616_nat_o(Xaa,produc7507926704131184380et_nat(H,As22)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_7786_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : ( linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_7787_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : ( linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ) ).

% sorted_list_of_set_atMost_Suc
tff(fact_7788_upto_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: list_int] :
      ( ( upto(Xa,Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(Xa,Xaa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa),cons_int(Xa,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),one_one_int),Xaa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(Xa,Xaa)) ) ) ) ).

% upto.pelims
tff(fact_7789_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_7790_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_7791_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_7792_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_7793_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt_int(ord_less_int,upto(I,J)) ).

% sorted_wrt_upto
tff(fact_7794_sorted__upto,axiom,
    ! [M: int,Na: int] : sorted_wrt_int(ord_less_eq_int,upto(M,Na)) ).

% sorted_upto
tff(fact_7795_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_7796_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_7797_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ).

% upto.simps
tff(fact_7798_upto_Oelims,axiom,
    ! [Xa: int,Xaa: int,Y: list_int] :
      ( ( upto(Xa,Xaa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa),cons_int(Xa,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_7799_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_7800_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_7801_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_7802_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I,J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_7803_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Na: num] :
      ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),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,Na)))),nil_int) ) ).

% upto_rec_numeral(2)
tff(fact_7804_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Na: num] :
      ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)),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,Na))),nil_int) ) ).

% upto_rec_numeral(3)
tff(fact_7805_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Na: num] :
      ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),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,Na)))),nil_int) ) ).

% upto_rec_numeral(4)
tff(fact_7806_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_7807_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Na: num] :
      ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)),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,Na))),nil_int) ) ).

% upto_rec_numeral(1)
tff(fact_7808_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_7809_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_num_real(numeral_numeral_real,W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,W2)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_7810_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),aa_nat_real(semiri5074537144036343181t_real,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_7811_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Na: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),Na))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_7812_powr__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_num_real(numeral_numeral_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,Na)) ) ) ).

% powr_numeral
tff(fact_7813_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_7814_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_7815_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_7816_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xa: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xa)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xa)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_7817_powr__neg__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,Na))) ) ) ).

% powr_neg_numeral
tff(fact_7818_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(1)
tff(fact_7819_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(2)
tff(fact_7820_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na)
    <=> ( Na != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_7821_enat__less__induct,axiom,
    ! [P2: fun_Extended_enat_o,Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M2),N)
             => aa_Extended_enat_o(P2,M2) )
         => aa_Extended_enat_o(P2,N) )
     => aa_Extended_enat_o(P2,Na) ) ).

% enat_less_induct
tff(fact_7822_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Na),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_7823_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,Na))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_7824_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xa: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y)
     => ( plus_p3455044024723400733d_enat(Xa,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xa,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_7825_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Na),zero_z5237406670263579293d_enat)
    <=> ( Na = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_7826_i0__lb,axiom,
    ! [Na: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Na) ).

% i0_lb
tff(fact_7827_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( vEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_VEBT_insert(T),Xa)) = inf_inf_set_nat(sup_sup_set_nat(vEBT_set_vebt(T),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))) ) ) ).

% insert'_correct
tff(fact_7828_setceilmax,axiom,
    ! [Sa: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Sa),M)
     => ( ! [X: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Listy))
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Na) )
       => ( ( M = aa_nat_nat(suc,Na) )
         => ( ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Listy))
               => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) ) )
           => ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Sa)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(lattic8265883725875713057ax_nat,image_VEBT_VEBT_nat(vEBT_VEBT_height,aa_set3061394203342972348T_VEBT(insert_VEBT_VEBT(Sa),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_7829_pow__sum,axiom,
    ! [A: nat,B: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ).

% pow_sum
tff(fact_7830_mulcomm,axiom,
    ! [I: nat,Va2: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),I) ) ).

% mulcomm
tff(fact_7831_two__realpow__ge__two,axiom,
    ! [Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)) ) ).

% two_realpow_ge_two
tff(fact_7832_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xa: nat,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xa)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% member_bound
tff(fact_7833_valid__pres__insert,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Xa)),Na) ) ) ).

% valid_pres_insert
tff(fact_7834_misiz,axiom,
    ! [T: vEBT_VEBT,Na: nat,M: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( some_nat(M) = vEBT_vebt_mint(T) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% misiz
tff(fact_7835_insert__simp__mima,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        | ( Xa = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% insert_simp_mima
tff(fact_7836_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ).

% two_powr_height_bound_deg
tff(fact_7837_helpyd,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_succ(T,Xa) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% helpyd
tff(fact_7838_helpypredd,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( vEBT_vebt_pred(T,Xa) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% helpypredd
tff(fact_7839_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Xa)),Xa) ) ) ).

% valid_insert_both_member_options_add
tff(fact_7840_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),Xa)
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Y)),Xa) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_7841_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Xa)),Y)
           => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,T),Y)
              | ( Xa = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_7842_log__ceil__idem,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xa)))) ) ) ).

% log_ceil_idem
tff(fact_7843_count__buildup,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na))) ).

% count_buildup
tff(fact_7844_delt__out__of__range,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ).

% delt_out_of_range
tff(fact_7845_del__single__cont,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & ( Xa = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) ) ) ) ).

% del_single_cont
tff(fact_7846_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(78)
tff(fact_7847_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(71)
tff(fact_7848_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,M: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Na) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na) ) ) ).

% set_n_deg_not_0
tff(fact_7849_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,M),one) ).

% semiring_norm(75)
tff(fact_7850_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Na) ).

% semiring_norm(68)
tff(fact_7851_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList),Summary)),Xa)),one_one_nat) ) ).

% tdeletemimi'
tff(fact_7852_count__buildup_H,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))),aa_nat_real(semiri5074537144036343181t_real,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))),Na)))) ).

% count_buildup'
tff(fact_7853_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ).

% heigt_uplog_rel
tff(fact_7854_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_7855_succ__min,axiom,
    ! [Deg: nat,Xa: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),Xa) = some_nat(Mi) ) ) ) ).

% succ_min
tff(fact_7856_pred__max,axiom,
    ! [Deg: nat,Ma: nat,Xa: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa)
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),Xa) = some_nat(Ma) ) ) ) ).

% pred_max
tff(fact_7857_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)),one_one_real))) ) ).

% cnt_bound'
tff(fact_7858_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_7859_bit__concat__def,axiom,
    ! [H2: nat,L: nat,D: nat] : ( aa_nat_nat(vEBT_VEBT_bit_concat(H2,L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H2),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_7860_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_TBOUND_o(vEBT_V854960066525838166emberi(T,Tia,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% TBOUND_vebt_memberi
tff(fact_7861_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Deg) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Deg)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_7862_inrange,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(T)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))) ) ).

% inrange
tff(fact_7863_Suc__numeral,axiom,
    ! [Na: num] : ( aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Na),one)) ) ).

% Suc_numeral
tff(fact_7864_insert__correct,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( sup_sup_set_nat(vEBT_set_vebt(T),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) = vEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Xa)) ) ) ) ).

% insert_correct
tff(fact_7865_set__vebt__insert,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( vEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Xa)) = sup_sup_set_nat(vEBT_set_vebt(T),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) ) ) ) ).

% set_vebt_insert
tff(fact_7866_insert__corr,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
       => ( sup_sup_set_nat(vEBT_VEBT_set_vebt(T),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)) = vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Xa)) ) ) ) ).

% insert_corr
tff(fact_7867_zdiv__numeral__Bit0,axiom,
    ! [V2: num,W2: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W2)) ) ).

% zdiv_numeral_Bit0
tff(fact_7868_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Na)) ).

% semiring_norm(76)
tff(fact_7869_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),one) ).

% semiring_norm(69)
tff(fact_7870_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Na)
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Ma)
          & ? [M3: nat] :
              ( ( some_nat(M3) = vEBT_vebt_mint(Summary) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_7871_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ) ).

% add_2_eq_Suc'
tff(fact_7872_add__2__eq__Suc,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ) ).

% add_2_eq_Suc
tff(fact_7873_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_7874_div2__Suc__Suc,axiom,
    ! [M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),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,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div2_Suc_Suc
tff(fact_7875_mod2__Suc__Suc,axiom,
    ! [M: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),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))) ) ).

% mod2_Suc_Suc
tff(fact_7876_zmod__numeral__Bit0,axiom,
    ! [V2: num,W2: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W2))) ) ).

% zmod_numeral_Bit0
tff(fact_7877_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_7878_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_7879_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_7880_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_7881_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),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_7882_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_7883_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_7884_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_7885_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_7886_double__not__eq__Suc__double,axiom,
    ! [M: nat,Na: 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))),Na)) ) ).

% double_not_eq_Suc_double
tff(fact_7887_Suc__double__not__eq__double,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% Suc_double_not_eq_double
tff(fact_7888_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% pos2
tff(fact_7889_pos__mod__bound2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% pos_mod_bound2
tff(fact_7890_n__less__equal__power__2,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% n_less_equal_power_2
tff(fact_7891_le__num__One__iff,axiom,
    ! [Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),one)
    <=> ( Xa = one ) ) ).

% le_num_One_iff
tff(fact_7892_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_7893_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_7894_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_7895_nat__induct2,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( aa_nat_o(P2,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P2,N)
             => aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P2,Na) ) ) ) ).

% nat_induct2
tff(fact_7896_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),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),Na))) ) ).

% diff_le_diff_pow
tff(fact_7897_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_7898_realpow__square__minus__le,axiom,
    ! [U: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_7899_binomial__antimono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),K5)),aa_nat_nat(binomial(Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_7900_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_7901_pos__mod__sign2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% pos_mod_sign2
tff(fact_7902_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_7903_binomial__mono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K5)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),K5)) ) ) ).

% binomial_mono
tff(fact_7904_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_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))),Na)),K)),aa_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))),Na)),Na)) ).

% binomial_maximum'
tff(fact_7905_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)) ).

% two_realpow_ge_one
tff(fact_7906_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% binomial_le_pow2
tff(fact_7907_sin__gt__zero__02,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_gt_zero_02
tff(fact_7908_mod__exp__less__eq__exp,axiom,
    ! [A: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% mod_exp_less_eq_exp
tff(fact_7909_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_7910_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_7911_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_7912_nat__bit__induct,axiom,
    ! [P2: fun_nat_o,Na: nat] :
      ( aa_nat_o(P2,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P2,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P2,N)
             => aa_nat_o(P2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) )
         => aa_nat_o(P2,Na) ) ) ) ).

% nat_bit_induct
tff(fact_7913_two__pow__div__gt__le,axiom,
    ! [V2: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,V2),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))),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% two_pow_div_gt_le
tff(fact_7914_nat__add__offset__less,axiom,
    ! [Y: nat,Na: nat,Xa: nat,M: nat,Sz: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),Y)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_7915_nat__power__less__diff,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Q3),aa_nat_nat(aa_nat_fun_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,M),Na))) ) ).

% nat_power_less_diff
tff(fact_7916_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_7917_nat__le__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_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,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_le_power_trans
tff(fact_7918_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_7919_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K5)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),K5)) ) ) ).

% binomial_strict_mono
tff(fact_7920_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(binomial(Na),K5)),aa_nat_nat(binomial(Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_7921_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,Na))),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))),Na))) ).

% square_fact_le_2_fact
tff(fact_7922_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_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_7923_exp__plus__inverse__exp,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))) ).

% exp_plus_inverse_exp
tff(fact_7924_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList),Summary)),Xa) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_7925_triangle__def,axiom,
    ! [Na: nat] : ( aa_nat_nat(nat_triangle,Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ).

% triangle_def
tff(fact_7926_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_7927_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_7928_less__two__pow__divD,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),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))),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))) ) ) ).

% less_two_pow_divD
tff(fact_7929_less__two__pow__divI,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),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))),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ).

% less_two_pow_divI
tff(fact_7930_nat__less__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_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,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_less_power_trans
tff(fact_7931_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_7932_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C2: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_7933_plus__inverse__ge__2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(inverse_inverse_real,Xa))) ) ).

% plus_inverse_ge_2
tff(fact_7934_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_7935_Suc__nat__number__of__add,axiom,
    ! [V2: num,Na: nat] : ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V2)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V2),one))),Na) ) ).

% Suc_nat_number_of_add
tff(fact_7936_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_VEBT_VEBT(vEBT_VEBT_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)),Xa),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Xa)) ) ).

% VEBT_internal.insert'.simps(2)
tff(fact_7937_nat__div__eq__Suc__0__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_7938_power__2__mult__step__le,axiom,
    ! [N2: nat,Na: nat,K5: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),K5)),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))),Na)),K))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K5),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_7939_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_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_7940_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_7941_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_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_7942_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_7943_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_7944_exp__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_7945_sb__dec__lem_H,axiom,
    ! [K: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A)) ) ).

% sb_dec_lem'
tff(fact_7946_real__le__x__sinh,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_7947_real__le__abs__sinh,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xa)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_7948_mask__eq__sum__exp__nat,axiom,
    ! [Na: 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))),Na)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_br,Na))) ) ).

% mask_eq_sum_exp_nat
tff(fact_7949_gauss__sum__nat,axiom,
    ! [Na: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aq),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ).

% gauss_sum_nat
tff(fact_7950_sum__power2,axiom,
    ! [K: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ) ).

% sum_power2
tff(fact_7951_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_VEBT_insert(Xa),Xaa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( Y != aa_nat_VEBT_VEBT(vEBT_vebt_insert(vEBT_Leaf((A3),(B3))),Xaa) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)),Xaa),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)),Xaa)) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_7952_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_less
tff(fact_7953_sb__inc__lem,axiom,
    ! [A: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem
tff(fact_7954_sb__inc__lem_H,axiom,
    ! [A: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem'
tff(fact_7955_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_7956_sb__dec__lem,axiom,
    ! [K: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A)) ) ).

% sb_dec_lem
tff(fact_7957_ln__one__plus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_7958_real__exp__bound__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa))) ) ) ).

% real_exp_bound_lemma
tff(fact_7959_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),product_Pair_int_int(Q3,R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),product_Pair_int_int(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))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_7960_arith__series__nat,axiom,
    ! [A: nat,D: nat,Na: nat] : ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xa(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Na)) = 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,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ).

% arith_series_nat
tff(fact_7961_sinh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xa),aa_real_real(inverse_inverse_real,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_7962_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_7963_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_7964_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),Na)
     => ( groups4538972089207619220nt_int(aTP_Lamm_ar,set_or1266510415728281911st_int(M,Na)) = 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,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),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_7965_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_le
tff(fact_7966_exp__lower__Taylor__quadratic,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xa)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_7967_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_7968_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_7969_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Na) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_7970_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),product_Pair_int_int(Q3,R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),product_Pair_int_int(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))),R)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_7971_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_nat(binomial(Na),K) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K),
          zero_zero_nat,
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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(binomial(Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_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,Na),K)),one_one_nat),Na),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ) ).

% binomial_code
tff(fact_7972_floor__log__nat__eq__if,axiom,
    ! [B: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_7973_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Na) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Na) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_7974_arctan__double,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,Xa)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_7975_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Na)),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,Na)))),aa_nat_real(semiri5074537144036343181t_real,aa_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))),Na)),Na))) ) ).

% central_binomial_lower_bound
tff(fact_7976_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Na) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_7977_ln__one__minus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),Xa))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_7978_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_7979_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_7980_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_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,Na),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_7981_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_7982_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa))),Xa))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_7983_Tb__T__vebt__buildupi,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Na))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi
tff(fact_7984_Tb__T__vebt__buildupi_H,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(vEBT_V9176841429113362141ildupi,Na)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi'
tff(fact_7985_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_memberi(Tia,Xa),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_dq(T),Tia),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,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(bit1,aa_num_num(bit0,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_7986_Tb__T__vebt__buildupi_H_H,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(vEBT_VEBT_Tb2,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi''
tff(fact_7987_htt__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_predi(Tia,Xa),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_ds(T),Tia),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% htt_vebt_predi
tff(fact_7988_Tb__Tb_H,axiom,
    ! [T: nat] : ( aa_nat_int(vEBT_VEBT_Tb,T) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_VEBT_Tb2,T)) ) ).

% Tb_Tb'
tff(fact_7989_T__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% T_vebt_buildupi_univ
tff(fact_7990_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Mi)),Deg,TreeList),Summary)),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% tdeletemimi
tff(fact_7991_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(73)
tff(fact_7992_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(80)
tff(fact_7993_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ) ).

% minNull_delete_time_bound
tff(fact_7994_Tb_H__cnt,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_VEBT_Tb2,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na)))) ).

% Tb'_cnt
tff(fact_7995_TBOUND__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_VEBT_vebt_predi(T,Tia,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% TBOUND_vebt_predi
tff(fact_7996_TBOUND__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_VEBT_vebt_succi(T,Tia,Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% TBOUND_vebt_succi
tff(fact_7997_T__vebt__buildupi__cnt_H,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V441764108873111860ildupi,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na)))) ).

% T_vebt_buildupi_cnt'
tff(fact_7998_TBOUND__buildupi,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_vebt_buildupi(Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,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))),Na))) ) ).

% TBOUND_buildupi
tff(fact_7999_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ) ).

% delete_bound_height
tff(fact_8000_cnt__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)),aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))))) ) ).

% cnt_bound
tff(fact_8001_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(T,Tia,Xa)),
        $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))))) ).

% TBOUND_vebt_inserti
tff(fact_8002_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi_univ
tff(fact_8003_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi'_univ
tff(fact_8004_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_8005_time__vebt__delete,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)))))) ) ).

% time_vebt_delete
tff(fact_8006_time__vebt__pred,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)))))) ) ).

% time_vebt_pred
tff(fact_8007_time__vebt__succ,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)))))) ) ).

% time_vebt_succ
tff(fact_8008_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_inserti(Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Xa))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% htt_vebt_inserti
tff(fact_8009_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(81)
tff(fact_8010_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(72)
tff(fact_8011_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Na)) ).

% semiring_norm(77)
tff(fact_8012_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_8013_TBOUND__vebt__deletei,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V1365221501068881998eletei(T,Tia,Xa)),
        $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa)),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))))) ).

% TBOUND_vebt_deletei
tff(fact_8014_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_memberi(Tia,Xa),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_dq(T),Tia),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,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(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ).

% htt_vebt_memberi
tff(fact_8015_zdiv__numeral__Bit1,axiom,
    ! [V2: num,W2: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W2)) ) ).

% zdiv_numeral_Bit1
tff(fact_8016_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(79)
tff(fact_8017_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(74)
tff(fact_8018_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_inserti(Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(T),Xa))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_8019_htt__vebt__deletei,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_deletei(Tia,Xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(T),Xa))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,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,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% htt_vebt_deletei
tff(fact_8020_htt__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,T),Tia),vEBT_vebt_succi(Tia,Xa),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_dr(T),Tia),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% htt_vebt_succi
tff(fact_8021_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V2: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V2)) ) ).

% Suc_div_eq_add3_div_numeral
tff(fact_8022_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na)) ) ).

% div_Suc_eq_div_add3
tff(fact_8023_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V2: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V2)) ) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_8024_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na)) ) ).

% mod_Suc_eq_mod_add3
tff(fact_8025_zmod__numeral__Bit1,axiom,
    ! [V2: num,W2: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V2)),aa_num_int(numeral_numeral_int,W2)))),one_one_int) ) ).

% zmod_numeral_Bit1
tff(fact_8026_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_8027_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_8028_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : ( aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))) ) ).

% eval_nat_numeral(3)
tff(fact_8029_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_8030_Suc3__eq__add__3,axiom,
    ! [Na: nat] : ( aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na) ) ).

% Suc3_eq_add_3
tff(fact_8031_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% maxt_bound
tff(fact_8032_VEBT__internal_OTb_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,zero_zero_nat) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_8033_mint__bound,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% mint_bound
tff(fact_8034_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_8035_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Na) ) ).

% Suc_div_eq_add3_div
tff(fact_8036_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Na) ) ).

% Suc_mod_eq_add3_mod
tff(fact_8037_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% exp_le
tff(fact_8038_VEBT__internal_OTb_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,zero_zero_nat)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_8039_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_8040_insersimp,axiom,
    ! [T: vEBT_VEBT,Na: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,T),X_1)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insersimp
tff(fact_8041_insertsimp,axiom,
    ! [T: vEBT_VEBT,Na: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,T)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insertsimp
tff(fact_8042_small__powers__of__2,axiom,
    ! [Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),one_one_nat))) ) ).

% small_powers_of_2
tff(fact_8043_log__base__10__eq2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xa)) ) ) ).

% log_base_10_eq2
tff(fact_8044_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))) ) ).

% pred_bound_height
tff(fact_8045_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))) ) ).

% insert_bound_height
tff(fact_8046_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))) ) ).

% succ_bound_height
tff(fact_8047_log__base__10__eq1,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xa)) ) ) ).

% log_base_10_eq1
tff(fact_8048_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_8049_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_8050_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_8051_VEBT__internal_Ovebt__buildupi__rule,axiom,
    ! [Na: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_o_assn(pure_assn,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)),vEBT_vebt_buildupi(Na),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,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))),Na))) ).

% VEBT_internal.vebt_buildupi_rule
tff(fact_8052_vebt__buildupi__rule,axiom,
    ! [Na: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_o_assn(pure_assn,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot_set_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,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))),Na))) ).

% vebt_buildupi_rule
tff(fact_8053_vebt__memberi__rule,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi,Xa: nat] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_memberi(Tia,Xa),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aa_set2974300032405964540o_assn(aTP_Lamm_xb(Na),Sa),Tia),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,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(bit1,aa_num_num(bit0,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% vebt_memberi_rule
tff(fact_8054_vebt__predi__rule,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi,Xa: nat,Y: nat] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_predi(Tia,Xa),aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_xc(Na),Sa),Tia),Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% vebt_predi_rule
tff(fact_8055_vebt__succi__rule,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi,Xa: nat,Y: nat] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_succi(Tia,Xa),aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_xd(Na),Sa),Tia),Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% vebt_succi_rule
tff(fact_8056_vebt__inserti__rule,axiom,
    ! [Xa: nat,Na: nat,Sa: set_nat,Tia: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_inserti(Tia,Xa),vEBT_Intf_vebt_assn(Na,sup_sup_set_nat(Sa,aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% vebt_inserti_rule
tff(fact_8057_space__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space_bound
tff(fact_8058_vebt__space__linear__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% vebt_space_linear_bound
tff(fact_8059_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)),one_one_real))) ) ).

% space_2_pow_bound
tff(fact_8060_space__space_H,axiom,
    ! [T: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,T)),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T)) ).

% space_space'
tff(fact_8061_space_H__bound,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space'_bound
tff(fact_8062_space__cnt,axiom,
    ! [T: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,T))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,T))) ).

% space_cnt
tff(fact_8063_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,vEBT_Leaf((A),(B))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) ).

% VEBT_internal.space'.simps(1)
tff(fact_8064_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,vEBT_Leaf((A),(B))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ).

% VEBT_internal.space.simps(1)
tff(fact_8065_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList)),zero_zero_nat)) ) ).

% VEBT_internal.space'.simps(2)
tff(fact_8066_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_8067_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList)),zero_zero_nat)) ) ).

% VEBT_internal.space.simps(2)
tff(fact_8068_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_8069_t__build__cnt,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_build_cnt
tff(fact_8070_t__buildup__cnt,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Na))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_buildup_cnt
tff(fact_8071_vebt__buildup__bound,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_8072_buildup__build__time,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Na)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Na)) ).

% buildup_build_time
tff(fact_8073_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_8074_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_8075_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_8076_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_8077_div__half__nat,axiom,
    ! [Y: nat,Xa: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y)) = $let(
            q: nat,
            q:= 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,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)),
            $let(
              r2: nat,
              r2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,q),Y)),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),r2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,q),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,r2),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(q),r2)) ) ) ) ) ).

% div_half_nat
tff(fact_8078_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),vEBT_Leaf((A3),(B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_8079_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xa) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),Xa)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),vEBT_Leaf((A3),(B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_8080_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_VEBT_insert(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = aa_nat_VEBT_VEBT(vEBT_vebt_insert(vEBT_Leaf((A3),(B3))),Xaa) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2) )
               => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)),Xaa),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2)),Xaa)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary2),Xaa)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_8081_pred__list__to__short,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),Xa) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_8082_high__def,axiom,
    ! [Xa: nat,Na: nat] : ( aa_nat_nat(vEBT_VEBT_high(Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ).

% high_def
tff(fact_8083_high__bound__aux,axiom,
    ! [Ma: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Ma),Na)),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_8084_high__inv,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_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))),Na))),Xa)),Na) = Y ) ) ).

% high_inv
tff(fact_8085_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),Xa) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_8086_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( set_or8404916559141939852nteger(L,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(U),one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_8087_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(K),zero_z3403309356797280102nteger) = K ) ).

% plus_integer_code(1)
tff(fact_8088_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(zero_z3403309356797280102nteger),L) = L ) ).

% plus_integer_code(2)
tff(fact_8089_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : ( image_4470545334726330049nteger(aTP_Lamm_xe(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(U),L))) = set_or8404916559141939852nteger(L,U) ) ).

% image_add_integer_atLeastLessThan
tff(fact_8090_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_8091_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(K),zero_z3403309356797280102nteger) = K ) ).

% minus_integer_code(1)
tff(fact_8092_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(zero_z3403309356797280102nteger),L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ) ).

% minus_integer_code(2)
tff(fact_8093_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_8094_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
tff(fact_8095_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
tff(fact_8096_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_8097_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = $ite(
          K = zero_z3403309356797280102nteger,
          zero_z3403309356797280102nteger,
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ) ).

% sgn_integer_code
tff(fact_8098_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),K) ) ).

% abs_integer_code
tff(fact_8099_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_8100_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_8101_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat,Va2: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Na)
     => ( ( Na = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Mi)
         => ( ( Ma != Mi )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList)) ) ) ) ) ).

% nested_mint
tff(fact_8102_insert__simp__norm,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xa != Ma )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xa),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary),aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_8103_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,Xa: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_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(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xa != Ma )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_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_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_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(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))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_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_VEBT_VEBT(vEBT_vebt_insert(Summary),aa_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_8104_bit__split__inv,axiom,
    ! [Xa: nat,D: nat] : ( aa_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(vEBT_VEBT_high(Xa),D),aa_nat_nat(vEBT_VEBT_low(Xa),D)),D) = Xa ) ).

% bit_split_inv
tff(fact_8105_low__def,axiom,
    ! [Xa: nat,Na: nat] : ( aa_nat_nat(vEBT_VEBT_low(Xa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ).

% low_def
tff(fact_8106_low__inv,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_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))),Na))),Xa)),Na) = Xa ) ) ).

% low_inv
tff(fact_8107_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg)
     => ( ( Mi != Ma )
       => ( the_nat(vEBT_vebt_maxt(Summary)) = aa_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_8108_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Xa) ) ) ) ).

% both_member_options_ding
tff(fact_8109_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_8110_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xa = Mi )
          | ( Xa = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_8111_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or189985376899183464nteger(L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_8112_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
        & ( ( Xa = Mi )
          | ( Xa = Ma )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_8113_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_8114_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H2 )
           => ( ( aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L) )
               => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H2,Newnode) )
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                     => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                  $ite(Xa = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H2)))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_8115_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_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)))) = H2 )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_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 )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_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(TreeList))
                 => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H2,Newnode) )
                     => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(Xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H2)))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_8116_del__x__not__mi,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H2 )
           => ( ( aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H2,Newnode) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                   => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H2),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                    $ite(
                                      Xa = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(maxs)))))) ),
                                      Ma ))),Deg,Newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                  $ite(Xa = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H2)))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_8117_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H2 )
           => ( ( aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L) )
               => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Sn = aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H2) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H2,Newnode) )
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                    $ite(
                                      Xa = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none_nat,Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(maxs)))))) ),
                                      Ma ))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_8118_del__x__not__mia,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H2 )
           => ( ( aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
               => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeList,H2,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H2),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                    $ite(
                                      Xa = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                      Ma ))),Deg,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),
                                  $ite(Xa = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),H2)))),Ma))),Deg,newlist),Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_8119_del__x__mia,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = $let(
                xn: nat,
                xn:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))),
                $let(
                  h: nat,
                  h:= aa_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)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_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))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),h),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                      Ma ))),Deg,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                  $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Ma))),Deg,newlist),Summary) ) ) ),
                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_8120_del__x__mi__lets__in__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_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)))) = H2 )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_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 )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_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(TreeList))
                 => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H2,Newnode) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( ( Sn = aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H2) )
                         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                      $ite(
                                        Xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(Sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(maxs)))))) ),
                                        Ma ))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_8121_del__x__mi__lets__in,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_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)))) = H2 )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_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 )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_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(TreeList))
                 => ( ( Newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H2,Newnode) )
                     => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H2),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                      $ite(
                                        Xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(maxs)))))) ),
                                        Ma ))),Deg,Newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(Xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H2)))),Ma))),Deg,Newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_8122_del__x__mi,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( Xa = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_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)))) = H2 )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))) )
             => ( ( aa_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 )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_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(TreeList))
                 => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),H2)),L),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeList,H2,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),H2),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                      $ite(
                                        Xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                        Ma ))),Deg,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(Xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),H2)))),Ma))),Deg,newlist),Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_8123_del__in__range,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa) = $let(
                xn: nat,
                xn:= 
                  $ite(Xa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))),Xa),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xa = Mi,xn,Mi),
                  $let(
                    h: nat,
                    h:= aa_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)))),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_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))))),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),h),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                      $ite(
                                        xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                        Ma ))),Deg,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                    $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Ma))),Deg,newlist),Summary) ) ) ),
                      aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_8124_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),Xa) = $let(
                l: nat,
                l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & vEBT_VEBT_less(some_nat(l),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summary,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_8125_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xa)
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),Xa) = $let(
              l: nat,
              l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & vEBT_VEBT_less(some_nat(l),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summary,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_8126_pred__lesseq__max,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma)
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),Xa) = $let(
              l: nat,
              l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & vEBT_VEBT_greater(some_nat(l),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa),some_nat(Mi),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_8127_pred__less__length__list,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary),Xa) = $let(
                l: nat,
                l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & vEBT_VEBT_greater(some_nat(l),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa),some_nat(Mi),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_8128_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_VEBT_low(Xa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_8129_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = $ite(
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) ),
          one_one_nat,
          $ite(
            ( ( Xa = Mi )
            & ( Xa = Ma ) ),
            one_one_nat,
            $let(
              xn: nat,
              xn:= 
                $ite(Xa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))),Xa),
              $let(
                l: nat,
                l:= aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),l)),
                      $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),l)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary),h),one_one_nat)),
                    one_one_nat ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_8130_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),Xa) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
          some_nat(Mi),
          $let(
            l: nat,
            l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                $let(
                  maxlow: option_nat,
                  maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                  $ite(
                    ( ( maxlow != none_nat )
                    & vEBT_VEBT_less(some_nat(l),maxlow) ),
                    aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h),l)),
                    $let(
                      sc: option_nat,
                      sc:= vEBT_vebt_succ(Summary,h),
                      $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(sc))))) ) ) ),
                none_nat ) ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_8131_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),Xa) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
          some_nat(Ma),
          $let(
            l: nat,
            l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                $let(
                  minlow: option_nat,
                  minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                  $ite(
                    ( ( minlow != none_nat )
                    & vEBT_VEBT_greater(some_nat(l),minlow) ),
                    aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h),l)),
                    $let(
                      pr: option_nat,
                      pr:= vEBT_vebt_pred(Summary,h),
                      $ite(
                        pr = none_nat,
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa),some_nat(Mi),none_nat),
                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(pr)))) ) ) ) ),
                none_nat ) ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_8132_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = $ite(
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) ),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),
          $ite(
            ( ( Xa = Mi )
            & ( Xa = Ma ) ),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary),
            $let(
              xn: nat,
              xn:= 
                $ite(Xa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))),Xa),
              $let(
                minn: nat,
                minn:= 
                  $ite(Xa = Mi,xn,Mi),
                $let(
                  h: nat,
                  h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),h),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                      Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                  $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),Summary) ) ) ),
                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_8133_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,Sa: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList),Sa)),Xa)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_8134_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $ite(
            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
            one_one_nat,
            $let(
              l: nat,
              l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                    $let(
                      minlow: option_nat,
                      minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                        $ite(
                          ( ( minlow != none_nat )
                          & vEBT_VEBT_greater(some_nat(l),minlow) ),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),l)),
                          $let(
                            pr: option_nat,
                            pr:= vEBT_vebt_pred(Summary,h),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary),h))),one_one_nat)),
                              $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(pr)))))) ) )) ),
                    one_one_nat )) ) ) )) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_8135_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $ite(
            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
            one_one_nat,
            $let(
              l: nat,
              l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)))),
                      $let(
                        maxlow: option_nat,
                        maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                          $ite(
                            ( ( maxlow != none_nat )
                            & vEBT_VEBT_less(some_nat(l),maxlow) ),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),l)),
                            $let(
                              sc: option_nat,
                              sc:= vEBT_vebt_succ(Summary,h),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary),h))),one_one_nat)),
                                $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(sc)))))) ) )) )),
                    one_one_nat )) ) ) )) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_8136_vebt__delete_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(Xa),Xaa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != vEBT_Leaf($false,(B3)) ) ) )
       => ( ! [A3: $o] :
              ( ? [B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf((A3),$false) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A3),(B3)) )
               => ( ? [N: nat] : ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( Y != vEBT_Leaf((A3),(B3)) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                   => ( Y != aa_VEB5832953974749077948T_VEBT(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] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                     => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi2,xn,Mi2),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                          $let(
                                            newlist: list_VEBT_VEBT,
                                            newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                            $ite(
                                              aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary2),h),
                                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                        $ite(
                                                          xn = Ma2,
                                                          $let(
                                                            maxs: option_nat,
                                                            maxs:= vEBT_vebt_maxt(sn),
                                                            $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                                          Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),sn) ),
                                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                      $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),Summary2) ) ) ),
                                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_8137_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Xa),Xaa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ? [N: nat] : ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                              one_one_nat,
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                one_one_nat,
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                          $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary2),h),one_one_nat)),
                                        one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_8138_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd2)),Xa)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_8139_vebt__succ_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( ! [Uu: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((Uu),(B3)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : ( Xa = vEBT_Leaf((Uv),(Uw)) )
           => ( ? [N: nat] : ( Xaa = aa_nat_nat(suc,N) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2) )
             => ( Y != none_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd) )
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            some_nat(Mi2),
                            $let(
                              l: nat,
                              l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & vEBT_VEBT_less(some_nat(l),maxlow) ),
                                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= vEBT_vebt_succ(Summary2,h),
                                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_8140_vebt__pred_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : ( Xa = vEBT_Leaf((Uu),(Uv)) )
         => ( ( Xaa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw: $o] : ( Xa = vEBT_Leaf((A3),(Uw)) )
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A3),(B3)) )
               => ( ? [Va: nat] : ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( Y != $ite(
                        (B3),
                        some_nat(one_one_nat),
                        $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3) )
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve) )
                 => ( Y != none_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              some_nat(Ma2),
                              $let(
                                l: nat,
                                l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & vEBT_VEBT_greater(some_nat(l),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= vEBT_vebt_pred(Summary2,h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa),some_nat(Mi2),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_8141_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa)
    <=> $ite(
          Xa = Mi,
          $true,
          $ite(
            Xa = Ma,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_8142_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList),Vc)),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_8143_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Xa),Xaa) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : ( Xa = vEBT_Leaf((Uu),(Uv)) )
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw: $o] : ( Xa = vEBT_Leaf((A3),(Uw)) )
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A3),(B3)) )
               => ( ? [Va: nat] : ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $ite((B3),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                one_one_nat,
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                            $ite(
                                              ( ( minlow != none_nat )
                                              & vEBT_VEBT_greater(some_nat(l),minlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                              $let(
                                                pr: option_nat,
                                                pr:= vEBT_vebt_pred(Summary2,h),
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary2),h))),one_one_nat)),
                                                  $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(pr)))))) ) )) ),
                                        one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_8144_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Xa),Xaa) = Y )
     => ( ( ? [Uu: $o,B3: $o] : ( Xa = vEBT_Leaf((Uu),(B3)) )
         => ( ( Xaa = zero_zero_nat )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : ( Xa = vEBT_Leaf((Uv),(Uw)) )
           => ( ? [N: nat] : ( Xaa = aa_nat_nat(suc,N) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                              one_one_nat,
                              $let(
                                l: nat,
                                l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),
                                        $let(
                                          maxlow: option_nat,
                                          maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                            $ite(
                                              ( ( maxlow != none_nat )
                                              & vEBT_VEBT_less(some_nat(l),maxlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                              $let(
                                                sc: option_nat,
                                                sc:= vEBT_vebt_succ(Summary2,h),
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary2),h))),one_one_nat)),
                                                  $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(sc)))))) ) )) )),
                                      one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_8145_vebt__member__code_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Xa)
    <=> $ite(
          ( ( Deg = zero_zero_nat )
          | ( Deg = aa_nat_nat(suc,zero_zero_nat) ) ),
          $false,
          $ite(
            Xa = Mi,
            $true,
            $ite(
              Xa = Ma,
              $true,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                $false,
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
                  $false,
                  $let(
                    h: nat,
                    h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% vebt_member_code(3)
tff(fact_8146_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi)
            | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa) ),
            one_one_nat,
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
              $ite(
                ( ( Xa = Mi )
                & ( Xa = Ma ) ),
                aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                          $ite(Xa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                  $let(
                    xn: nat,
                    xn:= 
                      $ite(Xa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),the_nat(vEBT_vebt_mint(Summary)))))),Xa),
                    $let(
                      l: nat,
                      l:= aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $let(
                        h: nat,
                        h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),l))),
                            $let(
                              newnode: vEBT_VEBT,
                              newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),l),
                              $let(
                                newlist: list_VEBT_VEBT,
                                newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                  $ite(
                                    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summary),h))),
                                      $let(
                                        sn: vEBT_VEBT,
                                        sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),h),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                          $ite(
                                            xn = Ma,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                              $let(
                                                maxs: option_nat,
                                                maxs:= vEBT_vebt_maxt(sn),
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                  $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) )),
                                            one_one_nat )) )),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                      $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h))),one_one_nat)) )) ) )),
                          one_one_nat ) ) ) )) )) )) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_8147_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xa),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw) )
           => (Y) )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3) )
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_8148_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xa),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3) )
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_8149_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xa),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw) )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3) )
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_8150_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xa),Xaa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
           => ~ ( ( Xaa = Mi2 )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2) )
             => ~ ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd) )
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_8151_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Xa),Xaa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ? [N: nat] : ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                                              $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary2)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $let(
                                            h: nat,
                                            h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                            $ite(
                                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l),
                                                  $let(
                                                    newlist: list_VEBT_VEBT,
                                                    newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                                      $ite(
                                                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summary2),h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary2),h),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                              $ite(
                                                                xn = Ma2,
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                  $let(
                                                                    maxs: option_nat,
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                      $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) )),
                                                                one_one_nat )) )),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                          $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h))),one_one_nat)) )) ) )),
                                              one_one_nat ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_8152_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Na) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),X_12)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma),Na) = I2 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(vEBT_VEBT_low(Ma),Na)) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(vEBT_VEBT_high(X),Na) = I2 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(vEBT_VEBT_low(X),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_8153_vebt__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xa),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
             => ~ $ite(
                    Xaa = Mi2,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_8154_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xa),Xaa)
     => ( ! [Uu: $o,Uv: $o] : ( Xa != vEBT_Leaf((Uu),(Uv)) )
       => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
               => ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2) )
                 => ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd) )
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_8155_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xa),Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : ( Xa = vEBT_Leaf((Uu),(Uv)) )
         => (Y) )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
           => (Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
               => ( (Y)
                <=> ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2) )
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd) )
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_8156_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $ite(
            Xa = Mi,
            zero_zero_nat,
            $ite(
              Xa = Ma,
              zero_zero_nat,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                zero_zero_nat,
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
                  zero_zero_nat,
                  $ite(
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xa)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Ma) ),
                    $let(
                      h: nat,
                      h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                    zero_zero_nat ) ) ) ) )) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_8157_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = $let(
          xn: nat,
          xn:= 
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Mi,Xa),
          $let(
            h: nat,
            h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
              & ~ ( ( Xa = Mi )
                  | ( Xa = Ma ) ) ),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary),h),one_one_nat)),
              one_one_nat ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_8158_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = $let(
          xn: nat,
          xn:= 
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Mi,Xa),
          $let(
            h: nat,
            h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
              & ~ ( ( Xa = Mi )
                  | ( Xa = Ma ) ) ),
              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Xa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary),h),Summary)),
              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary) ) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_8159_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),Na) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Na) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),X_12)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma),Na) = I2 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(vEBT_VEBT_low(Ma),Na)) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(vEBT_VEBT_high(X),Na) = I2 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I2)),aa_nat_nat(vEBT_VEBT_low(X),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList),Summary)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_8160_vebt__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xa),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
         => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2) )
           => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ( Xa != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_8161_vebt__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xa),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
           => (Y) )
         => ( ( ? [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2) )
             => (Y) )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
               => (Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_8162_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,Xa),Xaa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
           => ( Y != one_one_nat ) )
         => ( ( ? [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                          $ite(
                            Xaa = Mi2,
                            zero_zero_nat,
                            $ite(
                              Xaa = Ma2,
                              zero_zero_nat,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                zero_zero_nat,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  zero_zero_nat,
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa)
                                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma2) ),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                                    zero_zero_nat ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_8163_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
          $let(
            xn: nat,
            xn:= 
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),Mi,Xa),
            $let(
              h: nat,
              h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $ite(
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                & ~ ( ( Xa = Mi )
                    | ( Xa = Ma ) ) ),
                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary),h),one_one_nat)),
                one_one_nat ) ) )) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_8164_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Xa),Xaa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary2),h),one_one_nat)),
                              one_one_nat ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_8165_invar__vebt_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A12),A23)
    <=> ( ( ? [A6: $o,B6: $o] : ( A12 = vEBT_Leaf((A6),(B6)) )
          & ( A23 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList4: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT] :
            ( ( A12 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A23,TreeList4),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X5),N5) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),N5)
            & ( 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))),N5) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),N5) )
            & ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),X_12)
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4))
               => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X_12) ) )
        | ? [TreeList4: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT] :
            ( ( A12 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A23,TreeList4),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X5),N5) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N5))
            & ( 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,N5)) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),aa_nat_nat(suc,N5)) )
            & ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),X_12)
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4))
               => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X_12) ) )
        | ? [TreeList4: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A23,TreeList4),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X5),N5) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),N5)
            & ( 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))),N5) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),N5) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
               => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),X_12)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4))
                 => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A23))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
                 => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma3),N5) = I3 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),aa_nat_nat(vEBT_VEBT_low(Ma3),N5)) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(vEBT_VEBT_high(X5),N5) = I3 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),aa_nat_nat(vEBT_VEBT_low(X5),N5)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X5)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Ma3) ) ) ) ) ) )
        | ? [TreeList4: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A23,TreeList4),Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X5),N5) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N5))
            & ( 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,N5)) )
            & ( A23 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),aa_nat_nat(suc,N5)) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N5)))
               => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),X_12)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList4))
                 => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A23))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N5)))
                 => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma3),N5) = I3 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),aa_nat_nat(vEBT_VEBT_low(Ma3),N5)) )
                    & ! [X5: nat] :
                        ( ( ( aa_nat_nat(vEBT_VEBT_high(X5),N5) = I3 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList4),I3)),aa_nat_nat(vEBT_VEBT_low(X5),N5)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X5)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_8166_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A12),A23)
     => ( ( ? [A3: $o,B3: $o] : ( A12 = vEBT_Leaf((A3),(B3)) )
         => ( A23 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A12 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
             => ( ( A23 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N) )
                 => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M3)
                   => ( ( 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))),M3) )
                     => ( ( M3 = N )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M3) )
                         => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_13)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                                 => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A12 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
               => ( ( A23 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N) )
                   => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M3)
                     => ( ( 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))),M3) )
                       => ( ( M3 = aa_nat_nat(suc,N) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M3) )
                           => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_13)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                                   => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N) )
                     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M3)
                       => ( ( 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))),M3) )
                         => ( ( M3 = N )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M3) )
                             => ( ! [I4: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                   => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),X_12)
                                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I4) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                                       => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                             => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma2),N) = I4 )
                                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),aa_nat_nat(vEBT_VEBT_low(Ma2),N)) )
                                                & ! [X3: nat] :
                                                    ( ( ( aa_nat_nat(vEBT_VEBT_high(X3),N) = I4 )
                                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),aa_nat_nat(vEBT_VEBT_low(X3),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2),Summary2) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N) )
                       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M3)
                         => ( ( 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))),M3) )
                           => ( ( M3 = aa_nat_nat(suc,N) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M3) )
                               => ( ! [I4: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                     => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),X_12)
                                      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I4) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                                         => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                               => ( ( ( aa_nat_nat(vEBT_VEBT_high(Ma2),N) = I4 )
                                                   => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),aa_nat_nat(vEBT_VEBT_low(Ma2),N)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( aa_nat_nat(vEBT_VEBT_high(X3),N) = I4 )
                                                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I4)),aa_nat_nat(vEBT_VEBT_low(X3),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_8167_vebt__insert_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(Xa),Xaa) = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xa = vEBT_Leaf((A3),(B3)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
             => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
               => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) ) )
           => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Xaa,Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,h,aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary2),h),Summary2)),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_8168_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                $let(
                  minlow: option_nat,
                  minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                  $ite(
                    ( ( minlow != none_nat )
                    & vEBT_VEBT_greater(some_nat(l),minlow) ),
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),l)),
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary),h)),one_one_nat) ) ),
                one_one_nat ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_8169_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                $let(
                  maxlow: option_nat,
                  maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),
                  $ite(
                    ( ( maxlow != none_nat )
                    & vEBT_VEBT_less(some_nat(l),maxlow) ),
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),l)),
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary),h)),one_one_nat) ) ),
                one_one_nat ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_8170_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Xa),Xaa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2) )
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                            $let(
                              h: nat,
                              h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                & ~ ( ( Xaa = Mi2 )
                                    | ( Xaa = Ma2 ) ) ),
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),
                                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary2),h),one_one_nat)),
                                one_one_nat ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_8171_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Xa),Xaa) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : ( Xa = vEBT_Leaf((Uu),(Uv)) )
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw: $o] : ( Xa = vEBT_Leaf((A3),(Uw)) )
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ? [Va: nat] : ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              one_one_nat,
                              $let(
                                l: nat,
                                l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & vEBT_VEBT_greater(some_nat(l),minlow) ),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary2),h)),one_one_nat) ) ),
                                    one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_8172_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Xa),Xaa) = Y )
     => ( ( ? [Uu: $o,B3: $o] : ( Xa = vEBT_Leaf((Uu),(B3)) )
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : ( Xa = vEBT_Leaf((Uv),(Uw)) )
           => ( ? [N: nat] : ( Xaa = aa_nat_nat(suc,N) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            one_one_nat,
                            $let(
                              l: nat,
                              l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & vEBT_VEBT_less(some_nat(l),maxlow) ),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary2),h)),one_one_nat) ) ),
                                  one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_8173_in__children__def,axiom,
    ! [Na: nat,TreeList: list_VEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Na,TreeList),Xa)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),aa_nat_nat(vEBT_VEBT_high(Xa),Na))),aa_nat_nat(vEBT_VEBT_low(Xa),Na)) ) ).

% in_children_def
tff(fact_8174_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A3),(B3)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A3),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                    | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                      $ite(
                                        ( ( Xaa = Mi2 )
                                        & ( Xaa = Ma2 ) ),
                                        aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                                                  $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary2)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                              $let(
                                                h: nat,
                                                h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                                $ite(
                                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l),
                                                      $let(
                                                        newlist: list_VEBT_VEBT,
                                                        newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                                          $ite(
                                                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summary2),h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary2),h),
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                                  $ite(
                                                                    xn = Ma2,
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                      $let(
                                                                        maxs: option_nat,
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                          $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) )),
                                                                    one_one_nat )) )),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                              $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h))),one_one_nat)) )) ) )),
                                                  one_one_nat ) ) ) )) )) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_8175_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2),Xaa)) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd),Xaa)) ) )
               => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                  one_one_nat,
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                                        $ite(
                                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),
                                            $let(
                                              maxlow: option_nat,
                                              maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                                $ite(
                                                  ( ( maxlow != none_nat )
                                                  & vEBT_VEBT_less(some_nat(l),maxlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                                  $let(
                                                    sc: option_nat,
                                                    sc:= vEBT_vebt_succ(Summary2,h),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary2),h))),one_one_nat)),
                                                      $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(sc)))))) ) )) )),
                                          one_one_nat )) ) ) )) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_8176_vebt__succ_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = $ite((B3),some_nat(one_one_nat),none_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2),Xaa)) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd),Xaa)) ) )
               => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                some_nat(Mi2),
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & vEBT_VEBT_less(some_nat(l),maxlow) ),
                                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= vEBT_vebt_succ(Summary2,h),
                                            $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_8177_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((A3),(Uw)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A3),(B3)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite((B3),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3),Xaa)) ) )
               => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve),Xaa)) ) )
                 => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                    one_one_nat,
                                    $let(
                                      l: nat,
                                      l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $let(
                                        h: nat,
                                        h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            $let(
                                              minlow: option_nat,
                                              minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                                $ite(
                                                  ( ( minlow != none_nat )
                                                  & vEBT_VEBT_greater(some_nat(l),minlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                                  $let(
                                                    pr: option_nat,
                                                    pr:= vEBT_vebt_pred(Summary2,h),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary2),h))),one_one_nat)),
                                                      $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(pr)))))) ) )) ),
                                            one_one_nat )) ) ) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_8178_vebt__pred_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((A3),(Uw)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = $ite((A3),some_nat(zero_zero_nat),none_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A3),(B3)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = $ite(
                              (B3),
                              some_nat(one_one_nat),
                              $ite((A3),some_nat(zero_zero_nat),none_nat) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3),Xaa)) ) )
               => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve),Xaa)) ) )
                 => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  some_nat(Ma2),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & vEBT_VEBT_greater(some_nat(l),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= vEBT_vebt_pred(Summary2,h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa),some_nat(Mi2),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(pr)))) ) ) ) ),
                                        none_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_8179_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A3),(B3)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A3),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TreeList2),Summary2),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary2),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                  one_one_nat,
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    one_one_nat,
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary2),h),one_one_nat)),
                                            one_one_nat ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_8180_vebt__delete_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = vEBT_Leaf($false,(B3)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xa = vEBT_Leaf((A3),(B3)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf((A3),$false) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A3),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = vEBT_Leaf((A3),(B3)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary2),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                     => ( ( Y = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                       => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi2,xn,Mi2),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                              $let(
                                                newlist: list_VEBT_VEBT,
                                                newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                $ite(
                                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary2),h),
                                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                            $ite(
                                                              xn = Ma2,
                                                              $let(
                                                                maxs: option_nat,
                                                                maxs:= vEBT_vebt_maxt(sn),
                                                                $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                                              Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),sn) ),
                                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                          $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),Summary2) ) ) ),
                                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_8181_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz2),Xaa)) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd),Xaa)) ) )
               => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                one_one_nat,
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & vEBT_VEBT_less(some_nat(l),maxlow) ),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary2),h)),one_one_nat) ) ),
                                      one_one_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_8182_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xaa)) ) )
             => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                    & ~ ( ( Xaa = Mi2 )
                                        | ( Xaa = Ma2 ) ) ),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),
                                      $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary2),h),one_one_nat)),
                                    one_one_nat ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_8183_vebt__insert_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
               => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                 => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xaa)) ) )
             => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Xaa,Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList2,h,aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary2),h),Summary2)),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_8184_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((A3),(Uw)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xa = vEBT_Leaf((A3),(B3)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz2),Va3),Xaa)) ) )
               => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve),Xaa)) ) )
                 => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  one_one_nat,
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & vEBT_VEBT_greater(some_nat(l),minlow) ),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary2),h)),one_one_nat) ) ),
                                        one_one_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_8185_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xaa)) ) )
             => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList2),Summary2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary2),h),one_one_nat)),
                                  one_one_nat ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_8186_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),Xaa)) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2),Xaa)) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Mi2,
                                zero_zero_nat,
                                $ite(
                                  Xaa = Ma2,
                                  zero_zero_nat,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                    zero_zero_nat,
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                      zero_zero_nat,
                                      $ite(
                                        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa)
                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma2) ),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                                        zero_zero_nat ) ) ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_8187_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Xa),Xaa) = Y )
     => ( ( ? [A3: $o,B3: $o] : ( Xa = vEBT_Leaf((A3),(B3)) )
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
         => ( ( ? [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2) )
             => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] : ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $ite(
                            Xaa = Mi2,
                            one_one_nat,
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Ma2,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                                          $let(
                                            h: nat,
                                            h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_8188_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2),Uw2)),Xa) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_8189_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz)),Xa) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_8190_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xa: nat] : ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),Xa) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_8191_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,vEBT_Leaf((A),(B))),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          $ite(Xa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_8192_member__bound__height,axiom,
    ! [T: vEBT_VEBT,Na: nat,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,T),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% member_bound_height
tff(fact_8193_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,T),Na)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,T),Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_8194_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList),Summary)),Xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          $ite(
            Xa = Mi,
            one_one_nat,
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
              $ite(
                Xa = Ma,
                one_one_nat,
                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Mi),
                    one_one_nat,
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xa),
                        one_one_nat,
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(vEBT_VEBT_high(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),h)),aa_nat_nat(vEBT_VEBT_low(Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_8195_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Xa),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),Xaa)) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2) )
                 => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2),Xaa)) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              $ite(
                                Xaa = Mi2,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    Xaa = Ma2,
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                            one_one_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                                              $let(
                                                h: nat,
                                                h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_8196_vebt__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),Xaa)) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2),Xaa)) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi2,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_8197_vebt__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv),Uw),Xaa)) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz2),Xaa)) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2),Xaa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa))
                       => $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_8198_vebt__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList2),Summary2),Xaa))
                 => ~ $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_8199_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw),Xaa)) )
           => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_8200_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_8201_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xa = vEBT_Leaf((A3),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B3)),Xaa)) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw),Xaa)) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList2),S3),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_8202_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Xaa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2),Xaa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi2 )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2),Xaa)) ) )
               => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_8203_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Xaa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2),Xaa))
                   => ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2),Xaa))
                     => ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_8204_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(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(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va3),Vb2),Xaa))
               => ~ ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList2),Vc2),Xaa))
                 => ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList2),Vd),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(vEBT_VEBT_high(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(vEBT_VEBT_low(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_8205_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),Xa)
            & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ww,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8206_Maclaurin__sin__expansion4,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ? [T6: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),Xa)
          & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ww,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8207_Maclaurin__sin__expansion2,axiom,
    ! [Xa: real,Na: nat] :
    ? [T6: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(sin_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ww,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8208_sin__cos__npi,axiom,
    ! [Na: nat] : ( aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) ) ).

% sin_cos_npi
tff(fact_8209_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_8210_pi__not__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).

% pi_not_less_zero
tff(fact_8211_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_8212_sin__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_gt_zero
tff(fact_8213_sin__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_ge_zero
tff(fact_8214_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_8215_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_8216_sin__eq__0__pi,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),pi)
       => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
         => ( Xa = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8217_sin__zero__pi__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),pi)
     => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
      <=> ( Xa = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8218_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_8219_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_8220_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_8221_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_8222_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_8223_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_8224_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8225_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(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_8226_sin__gt__zero2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xa)) ) ) ).

% sin_gt_zero2
tff(fact_8227_sin__lt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xa)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_8228_sin__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xa) = aa_real_real(sin_real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_8229_sin__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_8230_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xa)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_8231_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_8232_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_8233_sin__le__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xa)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_8234_sin__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xa)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_8235_sin__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8236_sin__monotone__2pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xa)) ) ) ) ).

% sin_monotone_2pi
tff(fact_8237_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X) = Y )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% sin_total
tff(fact_8238_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8239_Maclaurin__cos__expansion2,axiom,
    ! [Xa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),Xa)
            & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xf,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8240_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),T6)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),zero_zero_real)
            & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xf,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8241_monoseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_xg,Xa)) ) ).

% monoseq_arctan_series
tff(fact_8242_cos__pi__eq__zero,axiom,
    ! [M: nat] : ( aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ) ).

% cos_pi_eq_zero
tff(fact_8243_cos__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),one_one_real) ).

% cos_le_one
tff(fact_8244_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_8245_cos__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_8246_cos__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( ( aa_real_real(cos_real,Xa) = aa_real_real(cos_real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_8247_cos__ge__minus__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,Xa)) ).

% cos_ge_minus_one
tff(fact_8248_abs__cos__le__one,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xa))),one_one_real) ).

% abs_cos_le_one
tff(fact_8249_cos__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8250_cos__monotone__0__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8251_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_8252_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_8253_cos__is__zero,axiom,
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X) = zero_zero_real )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X ) ) ) ).

% cos_is_zero
tff(fact_8254_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_8255_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8256_cos__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
            & ( aa_real_real(cos_real,X) = Y )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),pi)
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% cos_total
tff(fact_8257_sincos__principal__value,axiom,
    ! [Xa: real] :
    ? [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,Xa) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,Xa) ) ) ).

% sincos_principal_value
tff(fact_8258_monoseq__realpow,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xa)) ) ) ).

% monoseq_realpow
tff(fact_8259_sin__cos__le1,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xa)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xa)),aa_real_real(cos_real,Y))))),one_one_real) ).

% sin_cos_le1
tff(fact_8260_cos__double__less__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xa))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_8261_cos__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xa)) ) ) ).

% cos_gt_zero
tff(fact_8262_cos__gt__zero__pi,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xa)) ) ) ).

% cos_gt_zero_pi
tff(fact_8263_cos__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xa)) ) ) ).

% cos_ge_zero
tff(fact_8264_pi__half,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_xh) ).

% pi_half
tff(fact_8265_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_xh)) ).

% pi_def
tff(fact_8266_sincos__total__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),pi)
            & ( Xa = aa_real_real(cos_real,T6) )
            & ( Y = aa_real_real(sin_real,T6) ) ) ) ) ).

% sincos_total_pi
tff(fact_8267_sin__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ).

% sin_expansion_lemma
tff(fact_8268_cos__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ) ).

% cos_expansion_lemma
tff(fact_8269_sincos__total__pi__half,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T6: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xa = aa_real_real(cos_real,T6) )
              & ( Y = aa_real_real(sin_real,T6) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_8270_sincos__total__2pi__le,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T6: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xa = aa_real_real(cos_real,T6) )
          & ( Y = aa_real_real(sin_real,T6) ) ) ) ).

% sincos_total_2pi_le
tff(fact_8271_sincos__total__2pi,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T6),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xa = aa_real_real(cos_real,T6) )
               => ( Y != aa_real_real(sin_real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8272_Maclaurin__cos__expansion,axiom,
    ! [Xa: real,Na: nat] :
    ? [T6: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,Xa))
      & ( aa_real_real(cos_real,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xf,Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_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(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8273_lowi__hT,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_o(time_htt_nat(one_one_assn,vEBT_VEBT_lowi(Xa,Na),aa_nat_fun_nat_assn(aTP_Lamm_xi(Xa),Na)),one_one_nat) ).

% lowi_hT
tff(fact_8274_highi__hT,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_o(time_htt_nat(one_one_assn,vEBT_VEBT_highi(Xa,Na),aa_nat_fun_nat_assn(aTP_Lamm_xj(Xa),Na)),one_one_nat) ).

% highi_hT
tff(fact_8275_TBOUND__highi,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_o(time_TBOUND_nat(vEBT_VEBT_highi(Xa,Na)),one_one_nat) ).

% TBOUND_highi
tff(fact_8276_TBOUND__lowi,axiom,
    ! [Xa: nat,Na: nat] : aa_nat_o(time_TBOUND_nat(vEBT_VEBT_lowi(Xa,Na)),one_one_nat) ).

% TBOUND_lowi
tff(fact_8277_highi__h,axiom,
    ! [Xa: nat,Na: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_highi(Xa,Na),aa_nat_fun_nat_assn(aTP_Lamm_xj(Xa),Na)) ).

% highi_h
tff(fact_8278_lowi__h,axiom,
    ! [Xa: nat,Na: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_lowi(Xa,Na),aa_nat_fun_nat_assn(aTP_Lamm_xi(Xa),Na)) ).

% lowi_h
tff(fact_8279_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_8280_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_8281_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K)) ).

% set_bit_greater_eq
tff(fact_8282_set__encode__insert,axiom,
    ! [A4: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ~ aa_set_nat_o(member_nat(Na),A4)
       => ( aa_set_nat_nat(nat_set_encode,aa_set_nat_set_nat(insert_nat(Na),A4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),aa_set_nat_nat(nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_8283_sin__coeff__def,axiom,
    ! [X3: nat] :
      ( aa_nat_real(sin_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),zero_zero_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3))) ) ).

% sin_coeff_def
tff(fact_8284_ln__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xa) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_xk,Xa)) ) ) ) ).

% ln_series
tff(fact_8285_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_8286_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_8287_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_8288_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_8289_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_8290_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xl,Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_8291_even__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc
tff(fact_8292_even__Suc__Suc__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc_Suc_iff
tff(fact_8293_odd__Suc__div__two,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_8294_even__Suc__div__two,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_8295_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_8296_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ) ) ).

% even_diff_nat
tff(fact_8297_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_8298_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_8299_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat)
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_8300_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_8301_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
tff(fact_8302_gcd__nat_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_8303_dvd__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)
       => ( M = Na ) ) ) ).

% dvd_antisym
tff(fact_8304_dvd__pos__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_8305_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M) ) ) ).

% nat_dvd_not_less
tff(fact_8306_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_minus_self
tff(fact_8307_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_8308_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_8309_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_8310_set__encode__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_nat(nat_set_encode,A4) = aa_set_nat_nat(nat_set_encode,B4) )
        <=> ( A4 = B4 ) ) ) ) ).

% set_encode_eq
tff(fact_8311_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A4))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A4) ) ) ).

% even_set_encode_iff
tff(fact_8312_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% dvd_imp_le
tff(fact_8313_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_mult_cancel
tff(fact_8314_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_8315_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X: nat,Y3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_8316_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_8317_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),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,Na),Q3) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_8318_dvd__fact,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% dvd_fact
tff(fact_8319_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xl,M))) ) ).

% finite_divisors_nat
tff(fact_8320_gcd__nat_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_xm),zero_zero_nat) ).

% gcd_nat.ordering_top_axioms
tff(fact_8321_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_nat(nat_set_encode,A4) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_8322_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_8323_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_8324_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_8325_dvd__minus__add,axiom,
    ! [Q3: nat,Na: nat,R: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Q3))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_8326_diff__mod__le,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),D)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,D),B)) ) ) ).

% diff_mod_le
tff(fact_8327_mod__nat__eqI,axiom,
    ! [R: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_8328_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% odd_pos
tff(fact_8329_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_8330_central__binomial__odd,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(binomial(Na),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(binomial(Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8331_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M7)
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M7)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(aa_set7246812951530217472et_nat(image_nat_set_nat(aTP_Lamm_xn),M7))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_8332_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_8333_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),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_8334_Bernoulli__inequality__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xa))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xa)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_8335_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [Na: nat] :
      ( aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,Na),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(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,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,Na),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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_8336_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half)) ),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,half))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_8337_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,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(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,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,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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_8338_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [Na: nat] :
      ( aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),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(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),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(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_8339_VEBT__internal_OTb_H_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_8340_cos__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(cos_real,Xa) = zero_zero_real )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_8341_sin__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(sin_real,Xa) = zero_zero_real )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_8342_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [Na: nat] :
      ( aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ).

% VEBT_internal.Tb.simps(3)
tff(fact_8343_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Na: nat] :
      ( aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),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(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),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(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,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,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_8344_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
            $let(
              half: nat,
              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_8345_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va2))) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
            $let(
              half: nat,
              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_8346_vebt__buildup_Oelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_8347_VEBT__internal_OTb_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_8348_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_8349_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_8350_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != one_one_int ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_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,aa_num_num(bit0,one)))),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(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_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,aa_num_num(bit0,aa_num_num(bit0,one))))),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(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,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,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,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)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_8351_arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(arctan,Xa) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_xo,Xa)) ) ) ).

% arctan_series
tff(fact_8352_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = one_one_int )
               => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_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,aa_num_num(bit0,one)))),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(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_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,aa_num_num(bit0,aa_num_num(bit0,one))))),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(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,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,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_8353_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
             => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_8354_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_8355_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_8356_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),Na)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M) ) ) ).

% zdvd_not_zless
tff(fact_8357_abs__div,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),Xa)
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,Xa)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_8358_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_8359_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_8360_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_8361_int__div__sub__1,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),M)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),one_one_int)),M) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),M)),one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),M)) ) ) ).

% int_div_sub_1
tff(fact_8362_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_8363_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_8364_even__nat__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ) ).

% even_nat_iff
tff(fact_8365_eme1p,axiom,
    ! [Na: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( 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),Na)),D) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),D)) ) ) ) ) ).

% eme1p
tff(fact_8366_emep1,axiom,
    ! [Na: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int)),D) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),D)),one_one_int) ) ) ) ) ).

% emep1
tff(fact_8367_vebt__buildup_Opelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_8368_VEBT__internal_OTb_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_8369_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_8370_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xa) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
             => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat) ) )
         => ( ( ( Xa = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
               => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,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(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,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,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,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),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,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_8371_vebt__delete__code_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] : ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Xa) = case_o2442805151034396888at_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary),aa_nat6988169923125987140T_VEBT(aa_VEB5764018861059009915T_VEBT(aa_lis2657662162680668108T_VEBT(aa_nat7935216168507688033T_VEBT(aTP_Lamm_xq(Info),Deg),TreeList),Summary),Xa),Info) ) ).

% vebt_delete_code(2)
tff(fact_8372_divmod__int__def,axiom,
    ! [M: num,Na: num] : ( unique5052692396658037445od_int(M,Na) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))) ) ).

% divmod_int_def
tff(fact_8373_divmod_H__nat__def,axiom,
    ! [M: num,Na: num] : ( unique5055182867167087721od_nat(M,Na) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))) ) ).

% divmod'_nat_def
tff(fact_8374_vebt__insert__code_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary)),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Deg),one_one_nat),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary),case_o2442805151034396888at_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xa)),Deg,TreeList),Summary),produc3169358591047799142T_VEBT(aa_nat1426624462703894328T_VEBT(aa_VEB257809287009931607T_VEBT(aa_lis5232481317877780700T_VEBT(aTP_Lamm_xr(Deg),TreeList),Summary),Xa)),Info)) ) ).

% vebt_insert_code(2)
tff(fact_8375_minus__one__div__numeral,axiom,
    ! [Na: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Na))) ) ).

% minus_one_div_numeral
tff(fact_8376_one__div__minus__numeral,axiom,
    ! [Na: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Na))) ) ).

% one_div_minus_numeral
tff(fact_8377_numeral__div__minus__numeral,axiom,
    ! [M: num,Na: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Na))) ) ).

% numeral_div_minus_numeral
tff(fact_8378_minus__numeral__div__numeral,axiom,
    ! [M: num,Na: num] : ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Na))) ) ).

% minus_numeral_div_numeral
tff(fact_8379_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      ( divmod_nat(M,Na) = $ite(
          ( ( Na = zero_zero_nat )
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ),
          aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
          produc2626176000494625587at_nat(aTP_Lamm_xs,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na),Na)) ) ) ).

% divmod_nat_if
tff(fact_8380_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : ( unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_xt(L),Qr) ) ).

% divmod_step_integer_def
tff(fact_8381_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : ( unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_xu(L),Qr) ) ).

% divmod_step_nat_def
tff(fact_8382_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : ( unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_xv(L),Qr) ) ).

% divmod_step_int_def
tff(fact_8383_vebt__succ__code_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xa: nat] :
      ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Deg),one_one_nat),none_nat,case_o1383228350324149268at_nat(none_nat,produc2484365769952853102on_nat(aa_nat1972073180414880072on_nat(aa_VEB5583160467804174207on_nat(aa_lis5680555565288705232on_nat(aTP_Lamm_xw(Deg),TreeList),Summary),Xa)),Info)) ) ).

% vebt_succ_code(2)
tff(fact_8384_integer__of__int__code,axiom,
    ! [K: int] :
      ( aa_int_Code_integer(code_integer_of_int,K) = $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
          $ite(
            K = zero_zero_int,
            zero_z3403309356797280102nteger,
            $let(
              l: code_integer,
              l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(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))))),
              $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(l),one_one_Code_integer)) ) ) ) ) ).

% integer_of_int_code
tff(fact_8385_integer__of__int__eq__of__int,axiom,
    code_integer_of_int = ring_18347121197199848620nteger ).

% integer_of_int_eq_of_int
tff(fact_8386_divmod__integer_H__def,axiom,
    ! [M: num,Na: num] : ( unique3479559517661332726nteger(M,Na) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na))) ) ).

% divmod_integer'_def
tff(fact_8387_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_8388_less__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xb) ) ).

% less_integer.abs_eq
tff(fact_8389_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),Xb)) ) ).

% divide_integer.abs_eq
tff(fact_8390_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),Xb)) ) ).

% modulo_integer.abs_eq
tff(fact_8391_integer__of__int__inject,axiom,
    ! [Xa: int,Y: int] :
      ( aa_set_int_o(member_int(Xa),top_top_set_int)
     => ( aa_set_int_o(member_int(Y),top_top_set_int)
       => ( ( aa_int_Code_integer(code_integer_of_int,Xa) = aa_int_Code_integer(code_integer_of_int,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% integer_of_int_inject
tff(fact_8392_integer__of__int__induct,axiom,
    ! [P2: fun_Code_integer_o,Xa: code_integer] :
      ( ! [Y3: int] :
          ( aa_set_int_o(member_int(Y3),top_top_set_int)
         => aa_Code_integer_o(P2,aa_int_Code_integer(code_integer_of_int,Y3)) )
     => aa_Code_integer_o(P2,Xa) ) ).

% integer_of_int_induct
tff(fact_8393_integer__of__int__cases,axiom,
    ! [Xa: code_integer] :
      ~ ! [Y3: int] :
          ( ( Xa = aa_int_Code_integer(code_integer_of_int,Y3) )
         => ~ aa_set_int_o(member_int(Y3),top_top_set_int) ) ).

% integer_of_int_cases
tff(fact_8394_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_8395_Code__Numeral_Oset__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : ( aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(Xa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Xa),Xb)) ) ).

% Code_Numeral.set_bit_integer.abs_eq
tff(fact_8396_uminus__integer_Oabs__eq,axiom,
    ! [Xa: int] : ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,Xa)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,Xa)) ) ).

% uminus_integer.abs_eq
tff(fact_8397_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_8398_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa),Xb)) ) ).

% plus_integer.abs_eq
tff(fact_8399_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa),Xb)) ) ).

% minus_integer.abs_eq
tff(fact_8400_times__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xa),Xb)) ) ).

% times_integer.abs_eq
tff(fact_8401_abs__integer_Oabs__eq,axiom,
    ! [Xa: int] : ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(code_integer_of_int,Xa)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(abs_abs_int,Xa)) ) ).

% abs_integer.abs_eq
tff(fact_8402_sgn__integer_Oabs__eq,axiom,
    ! [Xa: int] : ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_int_Code_integer(code_integer_of_int,Xa)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(sgn_sgn_int,Xa)) ) ).

% sgn_integer.abs_eq
tff(fact_8403_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : ( aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% signed_take_bit_Suc_bit0
tff(fact_8404_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : ( aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_8405_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : ( aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% signed_take_bit_Suc_bit1
tff(fact_8406_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : ( aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_8407_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_8408_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_8409_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_8410_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_int_int(bit_ri631733984087533419it_int(Na),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_8411_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_8412_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_8413_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Na),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,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_8414_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_8415_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_8416_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na)))),aa_int_int(bit_ri631733984087533419it_int(Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_8417_lowi__def,axiom,
    ! [Xa: nat,Na: nat] : ( vEBT_VEBT_lowi(Xa,Na) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% lowi_def
tff(fact_8418_summable__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_xo,Xa)) ) ).

% summable_arctan_series
tff(fact_8419_highi__def,axiom,
    ! [Xa: nat,Na: nat] : ( vEBT_VEBT_highi(Xa,Na) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% highi_def
tff(fact_8420_highsimp,axiom,
    ! [Xa: nat,Na: nat] : ( heap_Time_return_nat(aa_nat_nat(vEBT_VEBT_high(Xa),Na)) = vEBT_VEBT_highi(Xa,Na) ) ).

% highsimp
tff(fact_8421_lowsimp,axiom,
    ! [Xa: nat,Na: nat] : ( heap_Time_return_nat(aa_nat_nat(vEBT_VEBT_low(Xa),Na)) = vEBT_VEBT_lowi(Xa,Na) ) ).

% lowsimp
tff(fact_8422_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N8: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_xx(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8423_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_xx(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_xx(F))) ) ).

% summable_rabs
tff(fact_8424_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_xy(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8425_suminf__eq__SUP__real,axiom,
    ! [X4: fun_nat_real] :
      ( summable_real(X4)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(X4,I2))
       => ( suminf_real(X4) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_xz(X4),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_8426_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_8427_tanh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_8428_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_8429_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_8430_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_8431_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_8432_tanh__real__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% tanh_real_less_iff
tff(fact_8433_tanh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xa)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% tanh_real_le_iff
tff(fact_8434_tanh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_8435_tanh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa) ) ).

% tanh_real_pos_iff
tff(fact_8436_tanh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% tanh_real_nonneg_iff
tff(fact_8437_tanh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_8438_flip__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : ( aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(Xa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Xa),Xb)) ) ).

% flip_bit_integer.abs_eq
tff(fact_8439_unset__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : ( aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(Xa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Xa),Xb)) ) ).

% unset_bit_integer.abs_eq
tff(fact_8440_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K)),K) ).

% unset_bit_less_eq
tff(fact_8441_tanh__real__lt__1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xa)),one_one_real) ).

% tanh_real_lt_1
tff(fact_8442_tanh__real__gt__neg1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,Xa)) ).

% tanh_real_gt_neg1
tff(fact_8443_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_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_8444_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X)) ) ) ).

% lemma_tan_total
tff(fact_8445_tan__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xa)) ) ) ).

% tan_gt_zero
tff(fact_8446_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X) = Y ) ) ).

% lemma_tan_total1
tff(fact_8447_tan__mono__lt__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8448_tan__monotone_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xa)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8449_tan__monotone,axiom,
    ! [Y: real,Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xa)) ) ) ) ).

% tan_monotone
tff(fact_8450_tan__total,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X) = Y )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X ) ) ) ).

% tan_total
tff(fact_8451_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X) = Y ) ) ) ).

% tan_total_pos
tff(fact_8452_tan__pos__pi2__le,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xa)) ) ) ).

% tan_pos_pi2_le
tff(fact_8453_tan__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xa)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_8454_tan__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8455_tan__mono__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xa)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_8456_tan__bound__pi2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,Xa))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_8457_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(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_8458_arctan__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xa)) = Xa ) ) ) ).

% arctan_tan
tff(fact_8459_arctan__unique,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xa) = Y )
         => ( aa_real_real(arctan,Y) = Xa ) ) ) ) ).

% arctan_unique
tff(fact_8460_tan__total__pi4,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z3) = Xa ) ) ) ).

% tan_total_pi4
tff(fact_8461_arctan__def,axiom,
    ! [Y: real] : ( aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_ya,Y)) ) ).

% arctan_def
tff(fact_8462_arcsin__def,axiom,
    ! [Y: real] : ( aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_yb,Y)) ) ).

% arcsin_def
tff(fact_8463_sin__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xa) = aa_real_real(divide_divide_real(aa_real_real(tan_real,Xa)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_8464_real__sqrt__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ).

% real_sqrt_less_iff
tff(fact_8465_real__sqrt__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ).

% real_sqrt_le_iff
tff(fact_8466_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_8467_real__sqrt__lt__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_8468_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_8469_real__sqrt__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_8470_real__sqrt__lt__1__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_8471_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_8472_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_8473_real__sqrt__le__1__iff,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_8474_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_8475_real__sqrt__pow2__iff,axiom,
    ! [Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa) ) ).

% real_sqrt_pow2_iff
tff(fact_8476_real__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa ) ) ).

% real_sqrt_pow2
tff(fact_8477_real__sqrt__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_8478_real__sqrt__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_8479_real__sqrt__ge__one,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_ge_one
tff(fact_8480_real__sqrt__eq__zero__cancel,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( ( aa_real_real(sqrt,Xa) = zero_zero_real )
       => ( Xa = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_8481_real__sqrt__ge__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_ge_zero
tff(fact_8482_real__sqrt__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xa)) ) ).

% real_sqrt_gt_zero
tff(fact_8483_real__div__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(divide_divide_real(Xa),aa_real_real(sqrt,Xa)) = aa_real_real(sqrt,Xa) ) ) ).

% real_div_sqrt
tff(fact_8484_sqrt__add__le__add__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xa)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_8485_le__real__sqrt__sumsq,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Xa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_8486_arcsin__le__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_8487_arcsin__minus,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xa)) ) ) ) ).

% arcsin_minus
tff(fact_8488_arcsin__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_8489_arcsin__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( ( aa_real_real(arcsin,Xa) = aa_real_real(arcsin,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_8490_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_8491_sqrt__divide__self__eq,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(divide_divide_real(aa_real_real(sqrt,Xa)),Xa) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa)) ) ) ).

% sqrt_divide_self_eq
tff(fact_8492_cos__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xa)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_8493_real__less__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_8494_real__le__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_8495_sqrt__le__D,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_8496_arcsin__less__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_8497_arcsin__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xa)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_8498_real__le__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xa)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_8499_real__sqrt__unique,axiom,
    ! [Y: real,Xa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xa )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,Xa) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_8500_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(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_8501_real__sqrt__sum__squares__ge1,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_8502_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_8503_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C2: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),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_8504_sqrt__ge__absD,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_8505_cos__arcsin__nonzero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xa)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_8506_real__less__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xa)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_8507_sqrt__sum__squares__le__sum,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_8508_real__inv__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xa) ) ) ).

% real_inv_sqrt_pow2
tff(fact_8509_real__sqrt__ge__abs1,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_8510_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_8511_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_8512_ln__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xa)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Xa)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_8513_real__sqrt__power__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xa)),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_8514_arsinh__real__aux,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_8515_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xa: real,Y: real,Xaa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_8516_arith__geo__mean__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xa),Y))),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_8517_powr__half__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(powr_real(Xa),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xa) ) ) ).

% powr_half_sqrt
tff(fact_8518_cos__x__y__le__one,axiom,
    ! [Xa: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Xa),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_8519_real__sqrt__sum__squares__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_8520_arcosh__real__def,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xa)
     => ( aa_real_real(arcosh_real,Xa) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_8521_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_8522_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_8523_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_8524_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_8525_sqrt__sum__squares__half__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_8526_arcsin__sin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xa)) = Xa ) ) ) ).

% arcsin_sin
tff(fact_8527_sin__cos__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xa))
     => ( aa_real_real(sin_real,Xa) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_8528_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(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_8529_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_8530_arcsin__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xa)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_8531_le__arcsin__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,Xa))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xa) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_8532_cos__tan,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xa) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_8533_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(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_8534_sin__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xa)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_8535_cos__arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_8536_arccos__le__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xa)) ) ) ) ).

% arccos_le_arccos
tff(fact_8537_arccos__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real) )
     => ( ( aa_real_real(arccos,Xa) = aa_real_real(arccos,Y) )
      <=> ( Xa = Y ) ) ) ).

% arccos_eq_iff
tff(fact_8538_arccos__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xa)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ) ).

% arccos_le_mono
tff(fact_8539_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_8540_arccos__less__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xa)) ) ) ) ).

% arccos_less_arccos
tff(fact_8541_arccos__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xa)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ) ).

% arccos_less_mono
tff(fact_8542_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_8543_arccos__cos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xa)) = Xa ) ) ) ).

% arccos_cos
tff(fact_8544_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_8545_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi)
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_8546_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_8547_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_8548_sin__arccos__nonzero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xa)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_8549_arccos__cos2,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xa)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xa)) = aa_real_real(uminus_uminus_real,Xa) ) ) ) ).

% arccos_cos2
tff(fact_8550_arccos__minus,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,Xa)) ) ) ) ).

% arccos_minus
tff(fact_8551_arccos__def,axiom,
    ! [Y: real] : ( aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_yc,Y)) ) ).

% arccos_def
tff(fact_8552_arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_8553_arccos__minus__abs,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xa)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,Xa)) ) ) ).

% arccos_minus_abs
tff(fact_8554_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_8555_cot__less__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xa)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_8556_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_yd),one_one_real) ).

% power_half_series
tff(fact_8557_cot__gt__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xa)) ) ) ).

% cot_gt_zero
tff(fact_8558_cosh__real__ge__1,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_ge_1
tff(fact_8559_cosh__real__nonneg,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_nonneg
tff(fact_8560_cosh__real__nonneg__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_8561_cosh__real__nonpos__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xa) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_8562_cosh__real__pos,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xa)) ).

% cosh_real_pos
tff(fact_8563_sinh__less__cosh__real,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xa)),aa_real_real(cosh_real,Xa)) ).

% sinh_less_cosh_real
tff(fact_8564_sinh__le__cosh__real,axiom,
    ! [Xa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xa)),aa_real_real(cosh_real,Xa)) ).

% sinh_le_cosh_real
tff(fact_8565_cosh__real__strict__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_8566_cosh__real__nonneg__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_8567_cosh__real__nonpos__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xa)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xa) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_8568_arcosh__cosh__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xa)) = Xa ) ) ).

% arcosh_cosh_real
tff(fact_8569_cosh__ln__real,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xa)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),aa_real_real(inverse_inverse_real,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8570_divmod__BitM__2__eq,axiom,
    ! [M: num] : ( unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),one_one_int) ) ).

% divmod_BitM_2_eq
tff(fact_8571_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : ( aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Na))) ) ).

% eval_nat_numeral(2)
tff(fact_8572_int__ge__less__than2__def,axiom,
    ! [D: int] : ( int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_ye(D))) ) ).

% int_ge_less_than2_def
tff(fact_8573_int__ge__less__than__def,axiom,
    ! [D: int] : ( int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_yf(D))) ) ).

% int_ge_less_than_def
tff(fact_8574_set__decode__0,axiom,
    ! [Xa: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xa))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xa) ) ).

% set_decode_0
tff(fact_8575_Suc__0__mod__eq,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_o_nat(zero_n2687167440665602831ol_nat,Na != aa_nat_nat(suc,zero_zero_nat)) ) ).

% Suc_0_mod_eq
tff(fact_8576_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_8577_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_nat_set_nat(nat_set_decode,aa_set_nat_nat(nat_set_encode,A4)) = A4 ) ) ).

% set_encode_inverse
tff(fact_8578_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R: int] : ( adjust_div(product_Pair_int_int(Q3,R)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q3),aa_o_int(zero_n2684676970156552555ol_int,R != zero_zero_int)) ) ).

% Divides.adjust_div_eq
tff(fact_8579_set__decode__Suc,axiom,
    ! [Na: nat,Xa: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Na)),aa_nat_set_nat(nat_set_decode,Xa))
    <=> aa_set_nat_o(member_nat(Na),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_8580_finite__set__decode,axiom,
    ! [Na: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(nat_set_decode,Na)) ).

% finite_set_decode
tff(fact_8581_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : ( adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_yg,Qr) ) ).

% Divides.adjust_div_def
tff(fact_8582_subset__decode__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% subset_decode_imp_le
tff(fact_8583_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_8584_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_8585_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: 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,Na))) = $ite(
          ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( Na = zero_zero_nat ) ),
          zero_zero_int,
          $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),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),Na)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)))))) ) ) ).

% divide_int_unfold
tff(fact_8586_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: 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,Na))) = $ite(
          ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
          | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
          | ( Na = zero_zero_nat ) ),
          aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)),
          $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))),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,Na),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))))) ) ) ).

% modulo_int_unfold
tff(fact_8587_False__map2__or,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Xs)),aa_set_o_set_o(insert_o($false),bot_bot_set_o))
     => ( ( size_size_list_o(Ys) = size_size_list_o(Xs) )
       => ( map_Pr7541730621154948341_o_o_o(produc6197397395684419436_o_o_o(fdisj),zip_o_o(Xs,Ys)) = Ys ) ) ) ).

% False_map2_or
tff(fact_8588_False__map2__and,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Xs)),aa_set_o_set_o(insert_o($false),bot_bot_set_o))
     => ( ( size_size_list_o(Ys) = size_size_list_o(Xs) )
       => ( map_Pr7541730621154948341_o_o_o(produc6197397395684419436_o_o_o(fconj),zip_o_o(Xs,Ys)) = Xs ) ) ) ).

% False_map2_and
tff(fact_8589_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ) ).

% take_bit_of_Suc_0
tff(fact_8590_UNIV__bool,axiom,
    top_top_set_o = aa_set_o_set_o(insert_o($false),aa_set_o_set_o(insert_o($true),bot_bot_set_o)) ).

% UNIV_bool
tff(fact_8591_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_8592_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_8593_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(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,Na),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_8594_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_8595_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ) ) ).

% take_bit_nat_eq
tff(fact_8596_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_8597_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_8598_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ).

% take_bit_nonnegative
tff(fact_8599_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_8600_take__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : ( aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(Xa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Xa),Xb)) ) ).

% take_bit_integer.abs_eq
tff(fact_8601_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(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))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_8602_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_nat_less_exp
tff(fact_8603_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_8604_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_int_less_exp
tff(fact_8605_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_8606_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ).

% take_bit_Suc_minus_bit0
tff(fact_8607_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_8608_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_8609_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_8610_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_8611_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),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))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_8612_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_8613_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),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))),Na)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_8614_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),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))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_8615_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ).

% take_bit_Suc_minus_bit1
tff(fact_8616_take__upt,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),Na)
     => ( take_nat(M,upt(I,Na)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_8617_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero_nat,J) = nil_nat )
    <=> ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_8618_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_8619_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K) )
     => ( append_nat(upt(I,J),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_8620_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_list_nat_nat(last_nat,upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_8621_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_8622_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_8623_sum__list__upt,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_list_nat_nat(groups4561878855575611511st_nat,upt(M,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_aq),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Na)) ) ) ).

% sum_list_upt
tff(fact_8624_upt__rec__numeral,axiom,
    ! [M: num,Na: num] :
      ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)),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,Na))),nil_nat) ) ).

% upt_rec_numeral
tff(fact_8625_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( upt(I,J) = append_nat(Xs,Ys) )
      <=> ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),J)
            & ( upt(I,K3) = Xs )
            & ( upt(K3,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_8626_sorted__upt,axiom,
    ! [M: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(M,Na)) ).

% sorted_upt
tff(fact_8627_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] : ( map_nat_nat(aTP_Lamm_yh(Ofs),upt(A,B)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),Ofs)) ) ).

% map_add_upt'
tff(fact_8628_map__Suc__upt,axiom,
    ! [M: nat,Na: nat] : ( map_nat_nat(suc,upt(M,Na)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Na)) ) ).

% map_Suc_upt
tff(fact_8629_map__add__upt,axiom,
    ! [Na: nat,M: nat] : ( map_nat_nat(aTP_Lamm_yh(Na),upt(zero_zero_nat,M)) = upt(Na,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ) ).

% map_add_upt
tff(fact_8630_sorted__wrt__upt,axiom,
    ! [M: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(M,Na)) ).

% sorted_wrt_upt
tff(fact_8631_upt__0,axiom,
    ! [I: nat] : ( upt(I,zero_zero_nat) = nil_nat ) ).

% upt_0
tff(fact_8632_butlast__upt,axiom,
    ! [M: nat,Na: nat] : ( butlast_nat(upt(M,Na)) = upt(M,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ).

% butlast_upt
tff(fact_8633_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Na: nat,Ns: list_nat,Q3: nat] :
      ( ( cons_nat(M,cons_nat(Na,Ns)) = upt(M,Q3) )
    <=> ( cons_nat(Na,Ns) = upt(aa_nat_nat(suc,M),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_8634_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_8635_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P2: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),U3)
     => ( ! [I2: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),I2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),U3) )
           => ~ aa_nat_o(P2,I2) )
       => ( filter_nat2(P2,upt(zero_zero_nat,U)) = filter_nat2(P2,upt(zero_zero_nat,U3)) ) ) ) ).

% upt_filter_extend
tff(fact_8636_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : ( map_nat_nat(aTP_Lamm_yi,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Na))) = upt(M,Na) ) ).

% map_decr_upt
tff(fact_8637_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( append_nat(upt(zero_zero_nat,I),upt(I,J)) = upt(zero_zero_nat,J) ) ) ).

% upt_append
tff(fact_8638_atLeastAtMost__upt,axiom,
    ! [Na: nat,M: nat] : ( aa_nat_set_nat(set_or1269000886237332187st_nat(Na),M) = aa_list_nat_set_nat(set_nat2,upt(Na,aa_nat_nat(suc,M))) ) ).

% atLeastAtMost_upt
tff(fact_8639_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_8640_atLeast__upt,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_ord_lessThan_nat,Na) = aa_list_nat_set_nat(set_nat2,upt(zero_zero_nat,Na)) ) ).

% atLeast_upt
tff(fact_8641_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xa: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xa,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xa )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_8642_atMost__upto,axiom,
    ! [Na: nat] : ( aa_nat_set_nat(set_ord_atMost_nat,Na) = aa_list_nat_set_nat(set_nat2,upt(zero_zero_nat,aa_nat_nat(suc,Na))) ) ).

% atMost_upto
tff(fact_8643_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ) ).

% upt_rec
tff(fact_8644_upt__eq__lel__conv,axiom,
    ! [L: nat,H2: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( upt(L,H2) = append_nat(Is1,cons_nat(I,Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa_nat_nat(suc,I),H2) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),H2) ) ) ).

% upt_eq_lel_conv
tff(fact_8645_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_8646_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( upt(I,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),append_nat(upt(I,J),cons_nat(J,nil_nat)),nil_nat) ) ).

% upt_Suc
tff(fact_8647_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : ( aa_int_int(bit_ri631733984087533419it_int(Na),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)))) ) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_8648_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(Na),K))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_8649_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Na),K)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_8650_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2)))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),Na) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_8651_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W2)))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),Na) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_8652_bin__nth__minus__Bit0,axiom,
    ! [Na: nat,W2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2))),Na)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% bin_nth_minus_Bit0
tff(fact_8653_bin__nth__minus__Bit1,axiom,
    ! [Na: nat,W2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W2))),Na)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% bin_nth_minus_Bit1
tff(fact_8654_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [Na: nat,K: int,L: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),L) )
     => ( map_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),upt(zero_zero_nat,Na)) = map_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),upt(zero_zero_nat,Na)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_8655_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_8656_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M2)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_8657_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ) ).

% take_bit_Suc_from_most
tff(fact_8658_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_8659_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_8660_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_8661_bit__integer_Oabs__eq,axiom,
    ! [Xa: int] : ( aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,aa_int_Code_integer(code_integer_of_int,Xa)) = aa_int_fun_nat_o(bit_se1146084159140164899it_int,Xa) ) ).

% bit_integer.abs_eq
tff(fact_8662_not__bit__Suc__0__numeral,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Na)) ).

% not_bit_Suc_0_numeral
tff(fact_8663_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Na)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ) ).

% bit_nat_iff
tff(fact_8664_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_8665_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_8666_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
tff(fact_8667_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : ( aa_int_int(bit_concat_bit(aa_nat_nat(suc,Na),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_concat_bit(Na,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ) ).

% concat_bit_Suc
tff(fact_8668_concat__bit__0,axiom,
    ! [K: int,L: int] : ( aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ) ).

% concat_bit_0
tff(fact_8669_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(Na,K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_8670_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_concat_bit(Na,K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_8671_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_concat_bit(M,K),L)),Na)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_8672_xor__Suc__0__eq,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Na),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,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% xor_Suc_0_eq
tff(fact_8673_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_8674_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_8675_xor__nat__numerals_I3_J,axiom,
    ! [Xa: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ) ).

% xor_nat_numerals(3)
tff(fact_8676_xor__nat__numerals_I4_J,axiom,
    ! [Xa: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa)) ) ).

% xor_nat_numerals(4)
tff(fact_8677_xor__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na) = $ite(
          M = zero_zero_nat,
          Na,
          $ite(Na = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ) ).

% xor_nat_unfold
tff(fact_8678_Suc__0__xor__eq,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% Suc_0_xor_eq
tff(fact_8679_or__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Na) = $ite(
          M = zero_zero_nat,
          Na,
          $ite(Na = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ) ).

% or_nat_unfold
tff(fact_8680_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_8681_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_8682_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_8683_or__nat__numerals_I4_J,axiom,
    ! [Xa: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ) ).

% or_nat_numerals(4)
tff(fact_8684_or__nat__numerals_I3_J,axiom,
    ! [Xa: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa)) ) ).

% or_nat_numerals(3)
tff(fact_8685_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_8686_xor__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : ( aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa),Xb)) ) ).

% xor_integer.abs_eq
tff(fact_8687_XOR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa),Y)) ) ) ).

% XOR_lower
tff(fact_8688_XOR__upper,axiom,
    ! [Xa: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xa),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% XOR_upper
tff(fact_8689_or__Suc__0__eq,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% or_Suc_0_eq
tff(fact_8690_Suc__0__or__eq,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ).

% Suc_0_or_eq
tff(fact_8691_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_8692_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_8693_OR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),Y)) ) ) ).

% OR_lower
tff(fact_8694_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)) ) ).

% or_greater_eq
tff(fact_8695_or__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : ( aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),Xb)) ) ).

% or_integer.abs_eq
tff(fact_8696_OR__upper,axiom,
    ! [Xa: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xa),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% OR_upper
tff(fact_8697_DERIV__even__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),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,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_8698_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H6)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H6))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_8699_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H6)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H6))),aa_real_real(F,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_8700_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F6: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F6,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
            & ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(minus_minus_real(B),A)),aa_real_real(F6,Z3)) ) ) ) ) ).

% MVT2
tff(fact_8701_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H6))),aa_real_real(F,Xa)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_8702_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xa),H6))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_8703_DERIV__ln,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xa),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_8704_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_8705_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_8706_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G3: fun_real_real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G3,X),topolo2177554685111907308n_real(X,top_top_set_real)) )
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_or1222579329274155063t_real(A,B))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G3,X)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_8707_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_8708_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_8709_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(minus_minus_real(Xa),H6))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_8710_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(minus_minus_real(Xa),H6))),aa_real_real(F,Xa)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_8711_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(minus_minus_real(Xa),H6)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(minus_minus_real(Xa),H6))),aa_real_real(F,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_8712_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xa: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(minus_minus_real(Xa),H6)),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xa)),aa_real_real(F,aa_real_real(minus_minus_real(Xa),H6))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_8713_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xa),Y3))),D)
             => ( aa_real_real(F,Xa) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_8714_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xa),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,Xa)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_8715_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xa: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xa),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xa)),aa_real_real(F,Y3)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_8716_DERIV__ln__divide,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(divide_divide_real(one_one_real),Xa),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_8717_DERIV__pow,axiom,
    ! [Na: nat,Xa: real,Sa: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_yj,Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xa,Sa)) ).

% DERIV_pow
tff(fact_8718_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aTP_Lamm_yk(R),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(powr_real(Z),aa_real_real(minus_minus_real(R),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_8719_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,Xa: real,R: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xa))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_yl(G),R),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(powr_real(aa_real_real(G,Xa)),aa_real_real(minus_minus_real(R),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_8720_DERIV__log,axiom,
    ! [Xa: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(log(B),aa_real_real(divide_divide_real(one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),Xa)),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_log
tff(fact_8721_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,Xa: real,F: fun_real_real,R: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xa,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xa))
       => ( has_fi5821293074295781190e_real(F,R,topolo2177554685111907308n_real(Xa,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_ym(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(powr_real(aa_real_real(G,Xa)),aa_real_real(F,Xa))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(ln_ln_real,aa_real_real(G,Xa)))),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F,Xa))),aa_real_real(G,Xa)))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_8722_DERIV__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_8723_DERIV__real__sqrt__generic,axiom,
    ! [Xa: real,D4: real] :
      ( ( Xa != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
         => ( D4 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
           => ( D4 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xa)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_8724_arcosh__real__has__field__derivative,axiom,
    ! [Xa: real,A4: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xa,A4)) ) ).

% arcosh_real_has_field_derivative
tff(fact_8725_artanh__real__has__field__derivative,axiom,
    ! [Xa: real,A4: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xa,A4)) ) ).

% artanh_real_has_field_derivative
tff(fact_8726_DERIV__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_8727_DERIV__arccos,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_8728_DERIV__arcsin,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_8729_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xa: real,Na: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X),topolo2177554685111907308n_real(X,top_top_set_real))
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yn(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_8730_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xa: real,Na: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M3: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X),topolo2177554685111907308n_real(X,top_top_set_real)) )
     => ? [T6: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,Xa))
          & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yn(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_8731_DERIV__odd__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( ( Xa != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_8732_Maclaurin,axiom,
    ! [H2: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T6: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H2) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),H2)
                & ( aa_real_real(F,H2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_yo(H2),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H2),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_8733_Maclaurin2,axiom,
    ! [H2: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T6: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ? [T6: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T6)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H2)
              & ( aa_real_real(F,H2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_yo(H2),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H2),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_8734_Maclaurin__minus,axiom,
    ! [H2: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T6: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H2),T6)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
           => ? [T6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),zero_zero_real)
                & ( aa_real_real(F,H2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_yo(H2),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H2),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_8735_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,Xa: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( Xa != zero_zero_real )
         => ( ! [M3: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X),topolo2177554685111907308n_real(X,top_top_set_real))
           => ? [T6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T6))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,Xa))
                & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yn(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_8736_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,Xa: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,T6: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,Xa)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
       => ? [T6: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T6)),aa_real_real(abs_abs_real,Xa))
            & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yn(Diff),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_8737_Taylor,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C2: real,Xa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T6: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xa)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),B)
                 => ( ( Xa != C2 )
                   => ? [T6: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),C2),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),T6)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),C2) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),T6)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),Xa) ) )
                        & ( aa_real_real(F,Xa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yp(Diff),C2),Xa),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xa),C2)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_8738_Taylor__up,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T6: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B)
             => ? [T6: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),T6)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),B)
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yq(Diff),B),C2),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(B),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_8739_Taylor__down,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T6: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T6)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B)
             => ? [T6: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T6)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T6),C2)
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yq(Diff),A),C2),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T6)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(A),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_8740_Maclaurin__lemma2,axiom,
    ! [Na: nat,H2: real,Diff: fun_na6724076748692969250l_real,K: nat,B4: real] :
      ( ! [M3: nat,T6: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H2) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T6),topolo2177554685111907308n_real(T6,top_top_set_real)) )
     => ( ( Na = aa_nat_nat(suc,K) )
       => ! [M2: nat,T7: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T7)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T7),H2) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_ys(Na),Diff),B4),M2),aa_real_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,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yt(Diff),M2),T7),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B4),aa_real_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,Na),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T7,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_8741_DERIV__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_yu,suminf_real(aa_real_fun_nat_real(aTP_Lamm_yv,Xa)),topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_8742_DERIV__real__root__generic,axiom,
    ! [Na: nat,Xa: real,D4: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( Xa != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xa)
             => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),zero_zero_real)
               => ( D4 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
               => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),D4,topolo2177554685111907308n_real(Xa,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_8743_DERIV__power__series_H,axiom,
    ! [R2: real,F: fun_nat_real,X0: real] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_yw(F),X)) )
     => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
         => has_fi5821293074295781190e_real(aTP_Lamm_yy(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_yw(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_8744_DERIV__isconst3,axiom,
    ! [A: real,B: real,Xa: real,Y: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_set_real_o(member_real(Xa),set_or1633881224788618240n_real(A,B))
       => ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(A,B))
         => ( ! [X: real] :
                ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X,top_top_set_real)) )
           => ( aa_real_real(F,Xa) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_8745_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F6: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_yz(F),N),aa_nat_real(aa_real_fun_nat_real(F6,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X: real] :
            ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X)) )
       => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F6,X0))
           => ( summable_real(L5)
             => ( ! [N: nat,X: real,Y3: real] :
                    ( aa_set_real_o(member_real(X),set_or1633881224788618240n_real(A,B))
                   => ( aa_set_real_o(member_real(Y3),set_or1633881224788618240n_real(A,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(F,X),N)),aa_nat_real(aa_real_fun_nat_real(F,Y3),N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N)),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y3)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_za(F),suminf_real(aa_real_fun_nat_real(F6,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_8746_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_int(code_int_of_integer,K) = $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
          aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
          $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_zb,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ) ).

% int_of_integer_code
tff(fact_8747_int__of__integer__max,axiom,
    ! [K: code_integer,L: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(ord_max_Code_integer(K),L)) = aa_int_int(ord_max_int(aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% int_of_integer_max
tff(fact_8748_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) ).

% finite_greaterThanLessThan
tff(fact_8749_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or4266950643985792945nteger(L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_8750_int__of__integer__of__nat,axiom,
    ! [Na: nat] : ( aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ) ).

% int_of_integer_of_nat
tff(fact_8751_integer__of__int__int__of__integer,axiom,
    ! [K: code_integer] : ( aa_int_Code_integer(code_integer_of_int,aa_Code_integer_int(code_int_of_integer,K)) = K ) ).

% integer_of_int_int_of_integer
tff(fact_8752_int__of__integer__integer__of__int,axiom,
    ! [K: int] : ( aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(code_integer_of_int,K)) = K ) ).

% int_of_integer_integer_of_int
tff(fact_8753_int__of__integer__inverse,axiom,
    ! [Xa: code_integer] : ( aa_int_Code_integer(code_integer_of_int,aa_Code_integer_int(code_int_of_integer,Xa)) = Xa ) ).

% int_of_integer_inverse
tff(fact_8754_of__int__integer__of,axiom,
    ! [K: code_integer] : ( aa_int_Code_integer(ring_18347121197199848620nteger,aa_Code_integer_int(code_int_of_integer,K)) = K ) ).

% of_int_integer_of
tff(fact_8755_int__of__integer__of__int,axiom,
    ! [K: int] : ( aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(ring_18347121197199848620nteger,K)) = K ) ).

% int_of_integer_of_int
tff(fact_8756_zero__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_8757_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ) ).

% card_greaterThanLessThan
tff(fact_8758_int__of__integer__numeral,axiom,
    ! [K: num] : ( aa_Code_integer_int(code_int_of_integer,aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_int(numeral_numeral_int,K) ) ).

% int_of_integer_numeral
tff(fact_8759_one__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_8760_uminus__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xa)) = aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,Xa)) ) ).

% uminus_integer.rep_eq
tff(fact_8761_plus__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% plus_integer.rep_eq
tff(fact_8762_minus__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% minus_integer.rep_eq
tff(fact_8763_times__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% times_integer.rep_eq
tff(fact_8764_abs__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xa)) = aa_int_int(abs_abs_int,aa_Code_integer_int(code_int_of_integer,Xa)) ) ).

% abs_integer.rep_eq
tff(fact_8765_divide__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% divide_integer.rep_eq
tff(fact_8766_modulo__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% modulo_integer.rep_eq
tff(fact_8767_sgn__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,Xa)) = aa_int_int(sgn_sgn_int,aa_Code_integer_int(code_int_of_integer,Xa)) ) ).

% sgn_integer.rep_eq
tff(fact_8768_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or5834768355832116004an_nat(L),U) ) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_8769_or__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% or_integer.rep_eq
tff(fact_8770_bit__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : ( aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,Xa) = aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_Code_integer_int(code_int_of_integer,Xa)) ) ).

% bit_integer.rep_eq
tff(fact_8771_int__of__integer__induct,axiom,
    ! [Y: int,P2: fun_int_o] :
      ( aa_set_int_o(member_int(Y),top_top_set_int)
     => ( ! [X: code_integer] : aa_int_o(P2,aa_Code_integer_int(code_int_of_integer,X))
       => aa_int_o(P2,Y) ) ) ).

% int_of_integer_induct
tff(fact_8772_int__of__integer__cases,axiom,
    ! [Y: int] :
      ( aa_set_int_o(member_int(Y),top_top_set_int)
     => ~ ! [X: code_integer] : ( Y != aa_Code_integer_int(code_int_of_integer,X) ) ) ).

% int_of_integer_cases
tff(fact_8773_int__of__integer,axiom,
    ! [Xa: code_integer] : aa_set_int_o(member_int(aa_Code_integer_int(code_int_of_integer,Xa)),top_top_set_int) ).

% int_of_integer
tff(fact_8774_type__definition__integer,axiom,
    aa_set_int_o(type_d8366093980585677751er_int(code_int_of_integer,code_integer_of_int),top_top_set_int) ).

% type_definition_integer
tff(fact_8775_integer__of__int__inverse,axiom,
    ! [Y: int] :
      ( aa_set_int_o(member_int(Y),top_top_set_int)
     => ( aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(code_integer_of_int,Y)) = Y ) ) ).

% integer_of_int_inverse
tff(fact_8776_int__of__integer__less__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y) ) ).

% int_of_integer_less_iff
tff(fact_8777_less__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% less_integer.rep_eq
tff(fact_8778_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_iff
tff(fact_8779_int__of__integer__inject,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ( aa_Code_integer_int(code_int_of_integer,Xa) = aa_Code_integer_int(code_int_of_integer,Y) )
    <=> ( Xa = Y ) ) ).

% int_of_integer_inject
tff(fact_8780_integer__eqI,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_int(code_int_of_integer,K) = aa_Code_integer_int(code_int_of_integer,L) )
     => ( K = L ) ) ).

% integer_eqI
tff(fact_8781_integer__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( K = L )
    <=> ( aa_Code_integer_int(code_int_of_integer,K) = aa_Code_integer_int(code_int_of_integer,L) ) ) ).

% integer_eq_iff
tff(fact_8782_less__eq__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_8783_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_eq_iff
tff(fact_8784_xor__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% xor_integer.rep_eq
tff(fact_8785_take__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_se1745604003318907178nteger(Xa),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Xa),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% take_bit_integer.rep_eq
tff(fact_8786_unset__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(Xa),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Xa),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% unset_bit_integer.rep_eq
tff(fact_8787_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(C2),X3))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_8788_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(C2),X3))),R3) )
               => ( aa_real_real(F,X3) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_8789_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(C2),X3))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X3)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_8790_Code__Numeral_Oset__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(Xa),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Xa),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% Code_Numeral.set_bit_integer.rep_eq
tff(fact_8791_flip__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(Xa),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Xa),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% flip_bit_integer.rep_eq
tff(fact_8792_greaterThanLessThan__upt,axiom,
    ! [Na: nat,M: nat] : ( aa_nat_set_nat(set_or5834768355832116004an_nat(Na),M) = aa_list_nat_set_nat(set_nat2,upt(aa_nat_nat(suc,Na),M)) ) ).

% greaterThanLessThan_upt
tff(fact_8793_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(L),one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_8794_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_8795_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_8796_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : ( code_divmod_integer(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),L),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(K),L)) ) ).

% divmod_integer_def
tff(fact_8797_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N7: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_zc(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_zc(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))),groups6591440286371151544t_real(aTP_Lamm_zc(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_8798_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N7: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_zc(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_zc(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_zc(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_8799_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_8800_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_8801_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => filterlim_nat_nat(aTP_Lamm_zd(C2),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_8802_monoseq__convergent,axiom,
    ! [X4: fun_nat_real,B4: real] :
      ( topolo6980174941875973593q_real(X4)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X4,I2))),B4)
       => ~ ! [L6: real] : ~ filterlim_nat_real(X4,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_8803_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ze(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L2: real] :
                ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N7)),L2)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L2),at_top_nat)
                & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),aa_nat_real(G,N7))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_8804_LIMSEQ__inverse__zero,axiom,
    ! [X4: fun_nat_real] :
      ( ! [R3: real] :
        ? [N8: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X4,N)) )
     => filterlim_nat_real(aTP_Lamm_zf(X4),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_8805_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_zg,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_8806_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zh,C2),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_8807_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zi,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_8808_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
       => ( ! [E: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
             => ? [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N7)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_8809_LIMSEQ__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_8810_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xa: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zj(Xa),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_8811_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C2)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_8812_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_8813_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xa)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zk,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_8814_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zl,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_8815_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zm,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_8816_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => summable_real(aTP_Lamm_zc(A)) ) ) ) ).

% summable
tff(fact_8817_zeroseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xa)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_xg,Xa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_8818_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_zc(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),suminf_real(aTP_Lamm_zc(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_8819_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_zn(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_zc(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_8820_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L2: real] :
              ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_zc(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)))),L2)
              & filterlim_nat_real(aTP_Lamm_zn(A),topolo2815343760600316023s_real(L2),at_top_nat)
              & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),groups6591440286371151544t_real(aTP_Lamm_zc(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_zo(A),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_8821_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_zc(A))),groups6591440286371151544t_real(aTP_Lamm_zc(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_8822_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_zo(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_zc(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_8823_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( code_num_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_zp,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% num_of_integer_code
tff(fact_8824_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_nat(code_nat_of_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_zq,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% nat_of_integer_code
tff(fact_8825_nat__of__integer__of__nat,axiom,
    ! [Na: nat] : ( aa_Code_integer_nat(code_nat_of_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = Na ) ).

% nat_of_integer_of_nat
tff(fact_8826_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_8827_of__nat__of__integer,axiom,
    ! [K: code_integer] : ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_Code_integer_nat(code_nat_of_integer,K)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(zero_z3403309356797280102nteger),K) ) ).

% of_nat_of_integer
tff(fact_8828_nat__of__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : ( aa_Code_integer_nat(code_nat_of_integer,Xa) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,Xa)) ) ).

% nat_of_integer.rep_eq
tff(fact_8829_nat__of__integer_Oabs__eq,axiom,
    ! [Xa: int] : ( aa_Code_integer_nat(code_nat_of_integer,aa_int_Code_integer(code_integer_of_int,Xa)) = aa_int_nat(nat2,Xa) ) ).

% nat_of_integer.abs_eq
tff(fact_8830_nat__of__integer__code__post_I1_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_8831_nat__of__integer__less__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(code_nat_of_integer,Xa)),aa_Code_integer_nat(code_nat_of_integer,Y))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_8832_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
     => ( set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U) = image_1215581382706833972nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_ord_lessThan_nat,aa_Code_integer_nat(code_nat_of_integer,U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_8833_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F: fun_real_real,F2: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zr(Na),F),at_bot_real,F2) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_8834_eventually__sequentially__Suc,axiom,
    ! [P2: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_wn(P2),at_top_nat)
    <=> eventually_nat(P2,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_8835_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_8836_eventually__sequentiallyI,axiom,
    ! [C2: nat,P2: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X)
         => aa_nat_o(P2,X) )
     => eventually_nat(P2,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_8837_eventually__sequentially,axiom,
    ! [P2: fun_nat_o] :
      ( eventually_nat(P2,at_top_nat)
    <=> ? [N9: nat] :
        ! [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N5)
         => aa_nat_o(P2,N5) ) ) ).

% eventually_sequentially
tff(fact_8838_le__sequentially,axiom,
    ! [F2: filter_nat] :
      ( ord_le2510731241096832064er_nat(F2,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N9),F2) ) ).

% le_sequentially
tff(fact_8839_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_zs(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_8840_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_8841_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_8842_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,A: real,B: real] :
      ( aa_set_real_o(ord_less_eq_set_real(image_nat_real(F,top_top_set_nat)),set_or1222579329274155063t_real(A,B))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_8843_Bseq__realpow,axiom,
    ! [Xa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xa),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_8844_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_8845_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F: fun_real_real,F2: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zr(Na),F),at_top_real,F2) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_8846_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) ).

% finite_greaterThanAtMost
tff(fact_8847_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_8848_greaterThanAtMost__upt,axiom,
    ! [Na: nat,M: nat] : ( aa_nat_set_nat(set_or6659071591806873216st_nat(Na),M) = aa_list_nat_set_nat(set_nat2,upt(aa_nat_nat(suc,Na),aa_nat_nat(suc,M))) ) ).

% greaterThanAtMost_upt
tff(fact_8849_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_8850_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_8851_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xa),Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : ( Xa = vEBT_Leaf((Uu),(Uv)) )
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ( (Y)
              <=> ~ ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                          ( ! [X5: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n2) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                          & case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X7)
                              & ! [X5: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                                 => ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zt(Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_8852_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xa),Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : ( Xa = vEBT_Leaf((Uu),(Uv)) )
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ~ ( ( Deg2 = Xaa )
                  & $let(
                      n2: nat,
                      n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m: nat,
                        m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                        ( ! [X5: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n2) )
                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                        & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                        & case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X7)
                            & ! [X5: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                               => ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zt(Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_8853_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or2715278749043346189nteger(L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_8854_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(L),one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_8855_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima2,Deg,TreeList),Summary)),Deg3)
    <=> ( ( Deg = Deg3 )
        & $let(
            n2: nat,
            n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m: nat,
              m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n2),
              ( ! [X5: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n2) )
              & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
              & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
              & case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X7)
                  & ! [X5: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList))
                     => ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zt(Deg),TreeList),Summary),n2),m)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_8856_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xa),Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : ( Xa = vEBT_Leaf((Uu),(Uv)) )
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
             => ( ( Deg2 = Xaa )
                & $let(
                    n2: nat,
                    n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m: nat,
                      m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                      ( ! [X5: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n2) )
                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                      & case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X7)
                          & ! [X5: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                             => ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zt(Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_8857_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Xaa))
                 => ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                          ( ! [X5: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n2) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                          & case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X7)
                              & ! [X5: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                                 => ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zt(Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_8858_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xa),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Xaa))
                 => ~ ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                            ( ! [X5: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n2) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                            & case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X7)
                                & ! [X5: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                                   => ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zt(Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_8859_Sup__real__def,axiom,
    ! [X4: set_real] : ( comple1385675409528146559p_real(X4) = ord_Least_real(aTP_Lamm_zu(X4)) ) ).

% Sup_real_def
tff(fact_8860_Sup__int__def,axiom,
    ! [X4: set_int] : ( complete_Sup_Sup_int(X4) = the_int(aTP_Lamm_zv(X4)) ) ).

% Sup_int_def
tff(fact_8861_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xa),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xa,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xa = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2) )
               => ( ( (Y)
                  <=> ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n2),
                            ( ! [X5: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X5),n2) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                            & case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X7)
                                & ! [X5: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,TreeList2))
                                   => ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zt(Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary2),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_8862_set__bits__int__def,axiom,
    ! [F: fun_nat_o] :
      ( bit_bi6516823479961619367ts_int(F) = $ite(
          ? [N5: nat] :
          ! [M5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),M5)
           => ( aa_nat_o(F,M5) = aa_nat_o(F,N5) ) ),
          $let(
            n2: nat,
            n2:= ord_Least_nat(aTP_Lamm_zw(F)),
            aa_int_int(bit_ri631733984087533419it_int(n2),groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),map_nat_o(F,upt(zero_zero_nat,aa_nat_nat(suc,n2))))) ),
          zero_zero_int ) ) ).

% set_bits_int_def
tff(fact_8863_set__bits__int__unfold_H,axiom,
    ! [F: fun_nat_o] :
      ( bit_bi6516823479961619367ts_int(F) = $ite(
          ? [N5: nat] :
          ! [N10: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N10)
           => ~ aa_nat_o(F,N10) ),
          groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),map_nat_o(F,upt(zero_zero_nat,ord_Least_nat(aTP_Lamm_zx(F))))),
          $ite(
            ? [N5: nat] :
            ! [N10: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N10)
             => aa_nat_o(F,N10) ),
            $let(
              n2: nat,
              n2:= ord_Least_nat(aTP_Lamm_zy(F)),
              aa_int_int(bit_ri631733984087533419it_int(n2),groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),append_o(map_nat_o(F,upt(zero_zero_nat,n2)),cons_o($true,nil_o)))) ),
            zero_zero_int ) ) ) ).

% set_bits_int_unfold'
tff(fact_8864_bin__last__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_bi6516823479961619367ts_int(F))
      <=> aa_nat_o(F,zero_zero_nat) ) ) ).

% bin_last_set_bits
tff(fact_8865_wf__set__bits__int__Suc,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(aTP_Lamm_wn(F))
    <=> bit_wf_set_bits_int(F) ) ).

% wf_set_bits_int_Suc
tff(fact_8866_wf__set__bits__int__simps,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
    <=> ? [N5: nat] :
          ( ! [N10: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N10)
             => ~ aa_nat_o(F,N10) )
          | ! [N10: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N10)
             => aa_nat_o(F,N10) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_8867_zeros,axiom,
    ! [Na: nat,F: fun_nat_o] :
      ( ! [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
         => ~ aa_nat_o(F,N4) )
     => bit_wf_set_bits_int(F) ) ).

% zeros
tff(fact_8868_wf__set__bits__int_Osimps,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
    <=> ( ? [N5: nat] :
          ! [N10: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N10)
           => ~ aa_nat_o(F,N10) )
        | ? [N5: nat] :
          ! [N10: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N10)
           => aa_nat_o(F,N10) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_8869_wf__set__bits__int_Ocases,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( ! [N: nat] :
            ~ ! [N3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
               => ~ aa_nat_o(F,N3) )
       => ~ ! [N: nat] :
              ~ ! [N3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N3)
                 => aa_nat_o(F,N3) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_8870_ones,axiom,
    ! [Na: nat,F: fun_nat_o] :
      ( ! [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N4)
         => aa_nat_o(F,N4) )
     => bit_wf_set_bits_int(F) ) ).

% ones
tff(fact_8871_int__set__bits__unfold__BIT,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( bit_bi6516823479961619367ts_int(F) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(F,zero_zero_nat))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_8872_bin__rest__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,bit_bi6516823479961619367ts_int(F)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_8873_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C7: $o] :
      ( ord_less_eq_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72)),char2((C0),(C1),(C22),(C32),(C42),(C52),(C62),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_zz,cons_o((B0),cons_o((B1),cons_o((B22),cons_o((B32),cons_o((B42),cons_o((B52),cons_o((B62),cons_o((B72),nil_o))))))))),zero_zero_nat)),aa_nat_nat(foldr_o_nat(aTP_Lamm_zz,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C42),cons_o((C52),cons_o((C62),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_eq_char_simp
tff(fact_8874_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C7: $o] :
      ( ord_less_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72)),char2((C0),(C1),(C22),(C32),(C42),(C52),(C62),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_zz,cons_o((B0),cons_o((B1),cons_o((B22),cons_o((B32),cons_o((B42),cons_o((B52),cons_o((B62),cons_o((B72),nil_o))))))))),zero_zero_nat)),aa_nat_nat(foldr_o_nat(aTP_Lamm_zz,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C42),cons_o((C52),cons_o((C62),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_char_simp
tff(fact_8875_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_8876_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ~ aa_set_nat_o(finite_finite_nat,aa_set_int_set_nat(image_int_nat(comp_int_nat_int(nat2,abs_abs_int)),S2)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_8877_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_8878_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( code_divmod_integer(K,L) = $ite(
          K = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
              $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_aaa(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_8879_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : ( code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) ).

% divmod_abs_code(6)
tff(fact_8880_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : ( code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,J)) ) ).

% divmod_abs_code(5)
tff(fact_8881_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( code_divmod_integer(K,L) = $ite(
          K = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
          $ite(
            aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
            $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_aab(L),code_divmod_abs(K,L))),
            $ite(
              L = zero_z3403309356797280102nteger,
              produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
              aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
                $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_aac(L),code_divmod_abs(K,L)))) ) ) ) ) ).

% divmod_integer_code
tff(fact_8882_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : ( code_divmod_abs(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))) ) ).

% divmod_abs_def
tff(fact_8883_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_aad(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% bit_cut_integer_code
tff(fact_8884_bit__cut__integer__def,axiom,
    ! [K: code_integer] : ( code_bit_cut_integer(K) = produc6677183202524767010eger_o(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),K)) ) ).

% bit_cut_integer_def
tff(fact_8885_len__num0,axiom,
    ! [Uu2: itself_Numeral_num0] : ( type_l4264026598287037464l_num0(Uu2) = zero_zero_nat ) ).

% len_num0
tff(fact_8886_smod__int__compares_I3_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(signed6292675348222524329lo_int(A),B)) ) ) ).

% smod_int_compares(3)
tff(fact_8887_smod__int__compares_I5_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(signed6292675348222524329lo_int(A),B)),aa_int_int(uminus_uminus_int,B)) ) ) ).

% smod_int_compares(5)
tff(fact_8888_smod__int__compares_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(signed6292675348222524329lo_int(A),B)),B) ) ) ).

% smod_int_compares(1)
tff(fact_8889_smod__int__compares_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(signed6292675348222524329lo_int(A),B)) ) ) ).

% smod_int_compares(2)
tff(fact_8890_smod__int__compares_I4_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(signed6292675348222524329lo_int(A),B)),zero_zero_int) ) ) ).

% smod_int_compares(4)
tff(fact_8891_smod__int__compares_I6_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(signed6292675348222524329lo_int(A),B)) ) ) ).

% smod_int_compares(6)
tff(fact_8892_smod__int__compares_I7_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(signed6292675348222524329lo_int(A),B)),zero_zero_int) ) ) ).

% smod_int_compares(7)
tff(fact_8893_smod__int__compares_I8_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(signed6292675348222524329lo_int(A),B)) ) ) ).

% smod_int_compares(8)
tff(fact_8894_smod__mod__positive,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_int(signed6292675348222524329lo_int(A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% smod_mod_positive
tff(fact_8895_min__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)) ) ).

% min_Suc_Suc
tff(fact_8896_min__0R,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),zero_zero_nat) = zero_zero_nat ) ).

% min_0R
tff(fact_8897_min__0L,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Na) = zero_zero_nat ) ).

% min_0L
tff(fact_8898_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B),aa_nat_nat(suc,A)) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(2)
tff(fact_8899_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A)),B) = aa_nat_nat(suc,A) ) ) ).

% min_Suc_gt(1)
tff(fact_8900_nat__mult__min__left,axiom,
    ! [M: nat,Na: 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),Na)),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,Na),Q3)) ) ).

% nat_mult_min_left
tff(fact_8901_nat__mult__min__right,axiom,
    ! [M: nat,Na: 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,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ) ).

% nat_mult_min_right
tff(fact_8902_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_8903_min__diff,axiom,
    ! [M: nat,I: nat,Na: 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,Na),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)),I) ) ).

% min_diff
tff(fact_8904_int__of__integer__min,axiom,
    ! [K: code_integer,L: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(ord_min_Code_integer(K),L)) = aa_int_int(ord_min_int(aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% int_of_integer_min
tff(fact_8905_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_8906_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_8907_drop__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ) ).

% drop_bit_Suc_minus_bit0
tff(fact_8908_drop__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ) ).

% drop_bit_Suc_minus_bit1
tff(fact_8909_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ) ).

% drop_bit_of_Suc_0
tff(fact_8910_drop__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : ( aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(Xa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Xa),Xb)) ) ).

% drop_bit_integer.abs_eq
tff(fact_8911_drop__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(Xa),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Xa),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% drop_bit_integer.rep_eq
tff(fact_8912_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% mask_nat_positive_iff
tff(fact_8913_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_8914_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_8915_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),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))),Na) ) ).

% push_bit_of_Suc_0
tff(fact_8916_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),K)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_8917_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_8918_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Na)) ).

% mask_nonnegative_int
tff(fact_8919_less__eq__mask,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ).

% less_eq_mask
tff(fact_8920_mask__integer_Orep__eq,axiom,
    ! [Xa: nat] : ( aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(bit_se2119862282449309892nteger,Xa)) = aa_nat_int(bit_se2000444600071755411sk_int,Xa) ) ).

% mask_integer.rep_eq
tff(fact_8921_push__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(Xa),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Xa),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% push_bit_integer.rep_eq
tff(fact_8922_mask__integer_Oabs__eq,axiom,
    ! [Xa: nat] : ( aa_nat_Code_integer(bit_se2119862282449309892nteger,Xa) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(bit_se2000444600071755411sk_int,Xa)) ) ).

% mask_integer.abs_eq
tff(fact_8923_push__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : ( aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(Xa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Xa),Xb)) ) ).

% push_bit_integer.abs_eq
tff(fact_8924_less__mask,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ) ).

% less_mask
tff(fact_8925_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q3)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_8926_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : ( aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% Suc_mask_eq_exp
tff(fact_8927_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% mask_nat_less_exp
tff(fact_8928_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_8929_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_8930_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_8931_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_8932_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K) ) ).

% and_less_eq
tff(fact_8933_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_8934_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_8935_AND__upper2,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)),Y) ) ).

% AND_upper2
tff(fact_8936_AND__upper1,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)),Xa) ) ).

% AND_upper1
tff(fact_8937_AND__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Y)) ) ).

% AND_lower
tff(fact_8938_drop__bit__int__code_I2_J,axiom,
    ! [Na: nat] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Na)),zero_zero_int) = zero_zero_int ) ).

% drop_bit_int_code(2)
tff(fact_8939_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_8940_and__nat__numerals_I3_J,axiom,
    ! [Xa: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xa))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ) ).

% and_nat_numerals(3)
tff(fact_8941_and__nat__numerals_I4_J,axiom,
    ! [Xa: num] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xa))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ) ).

% and_nat_numerals(4)
tff(fact_8942_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_8943_and__Suc__0__eq,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ).

% and_Suc_0_eq
tff(fact_8944_Suc__0__and__eq,axiom,
    ! [Na: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ).

% Suc_0_and_eq
tff(fact_8945_and__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(Xa),Xaa)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_Code_integer_int(code_int_of_integer,Xa)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% and_integer.rep_eq
tff(fact_8946_and__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : ( aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(aa_int_Code_integer(code_integer_of_int,Xa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xa),Xb)) ) ).

% and_integer.abs_eq
tff(fact_8947_and__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na) = $ite(
          ( ( M = zero_zero_nat )
          | ( Na = zero_zero_nat ) ),
          zero_zero_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% and_nat_unfold
tff(fact_8948_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,zero_zero_nat),I) = I ) ).

% drop_bit_int_code(1)
tff(fact_8949_push__bit__int__code_I1_J,axiom,
    ! [I: int] : ( aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,zero_zero_nat),I) = I ) ).

% push_bit_int_code(1)
tff(fact_8950_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_8951_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_8952_not__integer_Oabs__eq,axiom,
    ! [Xa: int] : ( aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_int_Code_integer(code_integer_of_int,Xa)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(bit_ri7919022796975470100ot_int,Xa)) ) ).

% not_integer.abs_eq
tff(fact_8953_not__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : ( aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,Xa)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_Code_integer_int(code_int_of_integer,Xa)) ) ).

% not_integer.rep_eq
tff(fact_8954_shiftl__Suc__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(bit_Sh3965577149348748681tl_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% shiftl_Suc_0
tff(fact_8955_shiftr__Suc__0,axiom,
    ! [Na: nat] : ( aa_nat_nat(bit_Sh2154871086232339855tr_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = aa_o_nat(zero_n2687167440665602831ol_nat,Na = zero_zero_nat) ) ).

% shiftr_Suc_0
tff(fact_8956_not__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_ri7919022796975470100ot_int),bit_ri7919022796975470100ot_int) ).

% not_integer.rsp
tff(fact_8957_xor__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se6526347334894502574or_int,bit_se6526347334894502574or_int) ).

% xor_integer.rsp
tff(fact_8958_less__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_8959_less__eq__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_eq_int,ord_less_eq_int) ).

% less_eq_integer.rsp
tff(fact_8960_minus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),minus_minus_int,minus_minus_int) ).

% minus_integer.rsp
tff(fact_8961_dup_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_aae),aTP_Lamm_aae) ).

% dup.rsp
tff(fact_8962_bit__integer_Orsp,axiom,
    bNF_re8347998229424194295_nat_o(fequal_int,fequal_fun_nat_o,bit_se1146084159140164899it_int,bit_se1146084159140164899it_int) ).

% bit_integer.rsp
tff(fact_8963_abs__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),abs_abs_int),abs_abs_int) ).

% abs_integer.rsp
tff(fact_8964_sgn__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),sgn_sgn_int),sgn_sgn_int) ).

% sgn_integer.rsp
tff(fact_8965_natural__of__integer_Orsp,axiom,
    bNF_re3715656647883201625at_nat(fequal_int,fequal_nat,nat2,nat2) ).

% natural_of_integer.rsp
tff(fact_8966_divide__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),divide_divide_int,divide_divide_int) ).

% divide_integer.rsp
tff(fact_8967_modulo__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),modulo_modulo_int,modulo_modulo_int) ).

% modulo_integer.rsp
tff(fact_8968_uminus__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),uminus_uminus_int),uminus_uminus_int) ).

% uminus_integer.rsp
tff(fact_8969_plus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),plus_plus_int,plus_plus_int) ).

% plus_integer.rsp
tff(fact_8970_times__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),times_times_int,times_times_int) ).

% times_integer.rsp
tff(fact_8971_or__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se1409905431419307370or_int,bit_se1409905431419307370or_int) ).

% or_integer.rsp
tff(fact_8972_and__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se725231765392027082nd_int,bit_se725231765392027082nd_int) ).

% and_integer.rsp
tff(fact_8973_xor__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se6528837805403552850or_nat,bit_se6528837805403552850or_nat) ).

% xor_natural.rsp
tff(fact_8974_drop__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se8568078237143864401it_int,bit_se8568078237143864401it_int) ).

% drop_bit_integer.rsp
tff(fact_8975_drop__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se8570568707652914677it_nat,bit_se8570568707652914677it_nat) ).

% drop_bit_natural.rsp
tff(fact_8976_take__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se2923211474154528505it_int,bit_se2923211474154528505it_int) ).

% take_bit_integer.rsp
tff(fact_8977_take__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se2925701944663578781it_nat,bit_se2925701944663578781it_nat) ).

% take_bit_natural.rsp
tff(fact_8978_less__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_8979_less__eq__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_8980_inj__on__diff__nat,axiom,
    ! [N6: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) )
     => aa_set_nat_o(inj_on_nat_nat(aTP_Lamm_vk(K)),N6) ) ).

% inj_on_diff_nat
tff(fact_8981_minus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),minus_minus_nat,minus_minus_nat) ).

% minus_natural.rsp
tff(fact_8982_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_aaf,aTP_Lamm_aaf) ).

% sub.rsp
tff(fact_8983_bit__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,fequal_fun_nat_o,bit_se1148574629649215175it_nat,bit_se1148574629649215175it_nat) ).

% bit_natural.rsp
tff(fact_8984_unset__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se4203085406695923979it_int,bit_se4203085406695923979it_int) ).

% unset_bit_integer.rsp
tff(fact_8985_flip__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se2159334234014336723it_int,bit_se2159334234014336723it_int) ).

% flip_bit_integer.rsp
tff(fact_8986_inj__Suc,axiom,
    ! [N6: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N6) ).

% inj_Suc
tff(fact_8987_Suc_Orsp,axiom,
    aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),suc),suc) ).

% Suc.rsp
tff(fact_8988_divide__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),divide_divide_nat,divide_divide_nat) ).

% divide_natural.rsp
tff(fact_8989_modulo__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),modulo_modulo_nat,modulo_modulo_nat) ).

% modulo_natural.rsp
tff(fact_8990_Code__Numeral_Oset__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se7879613467334960850it_int,bit_se7879613467334960850it_int) ).

% Code_Numeral.set_bit_integer.rsp
tff(fact_8991_plus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),plus_plus_nat,plus_plus_nat) ).

% plus_natural.rsp
tff(fact_8992_times__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),times_times_nat,times_times_nat) ).

% times_natural.rsp
tff(fact_8993_or__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se1412395901928357646or_nat,bit_se1412395901928357646or_nat) ).

% or_natural.rsp
tff(fact_8994_integer__of__natural_Orsp,axiom,
    bNF_re6650684261131312217nt_int(fequal_nat,fequal_int,semiri1314217659103216013at_int,semiri1314217659103216013at_int) ).

% integer_of_natural.rsp
tff(fact_8995_inj__on__set__encode,axiom,
    aa_set_set_nat_o(inj_on_set_nat_nat(nat_set_encode),collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_8996_mask__integer_Orsp,axiom,
    bNF_re6650684261131312217nt_int(fequal_nat,fequal_int,bit_se2000444600071755411sk_int,bit_se2000444600071755411sk_int) ).

% mask_integer.rsp
tff(fact_8997_mask__natural_Orsp,axiom,
    aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se2002935070580805687sk_nat),bit_se2002935070580805687sk_nat) ).

% mask_natural.rsp
tff(fact_8998_push__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se545348938243370406it_int,bit_se545348938243370406it_int) ).

% push_bit_integer.rsp
tff(fact_8999_push__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se547839408752420682it_nat,bit_se547839408752420682it_nat) ).

% push_bit_natural.rsp
tff(fact_9000_and__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se727722235901077358nd_nat,bit_se727722235901077358nd_nat) ).

% and_natural.rsp
tff(fact_9001_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_9002_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F)) ) ) ) ).

% suminf_reindex_mono
tff(fact_9003_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
         => ( ! [X: nat] :
                ( ~ aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(image_nat_nat(G),top_top_set_nat))
               => ( aa_nat_real(F,X) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_9004_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_wy,Na)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9005_unset__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se4205575877204974255it_nat,bit_se4205575877204974255it_nat) ).

% unset_bit_natural.rsp
tff(fact_9006_flip__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se2161824704523386999it_nat,bit_se2161824704523386999it_nat) ).

% flip_bit_natural.rsp
tff(fact_9007_set__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se7882103937844011126it_nat,bit_se7882103937844011126it_nat) ).

% set_bit_natural.rsp
tff(fact_9008_integer__of__num_I3_J,axiom,
    ! [Na: num] :
      ( aa_num_Code_integer(code_integer_of_num,aa_num_num(bit1,Na)) = $let(
          k: code_integer,
          k:= aa_num_Code_integer(code_integer_of_num,Na),
          aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(k),k)),one_one_Code_integer) ) ) ).

% integer_of_num(3)
tff(fact_9009_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
tff(fact_9010_integer__of__num__triv_I1_J,axiom,
    aa_num_Code_integer(code_integer_of_num,one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9011_integer__of__num_I2_J,axiom,
    ! [Na: num] :
      ( aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,Na)) = $let(
          k: code_integer,
          k:= aa_num_Code_integer(code_integer_of_num,Na),
          aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(k),k) ) ) ).

% integer_of_num(2)
tff(fact_9012_integer__of__num__triv_I2_J,axiom,
    aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,one)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9013_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_list_nat_nat(hd_nat,upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9014_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_9015_fst__divmod__nat,axiom,
    ! [M: nat,Na: nat] : ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(M,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) ) ).

% fst_divmod_nat
tff(fact_9016_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : ( produc8508995932063986495nteger(code_divmod_integer(K,L)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),L) ) ).

% fst_divmod_integer
tff(fact_9017_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : ( produc8508995932063986495nteger(code_divmod_abs(K,L)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)) ) ).

% fst_divmod_abs
tff(fact_9018_bezw_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xa,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_9019_bezw_Osimps,axiom,
    ! [Xa: nat,Y: nat] :
      ( bezw(Xa,Y) = $ite(Y = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y))),aa_int_int(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,Xa),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,Xa),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y)))))) ) ).

% bezw.simps
tff(fact_9020_bezw__non__0,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(Xa,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y))),aa_int_int(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,Xa),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,Xa),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Y))))) ) ) ).

% bezw_non_0
tff(fact_9021_bezw_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xa,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xa),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_9022_snd__divmod__nat,axiom,
    ! [M: nat,Na: nat] : ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,divmod_nat(M,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) ) ).

% snd_divmod_nat
tff(fact_9023_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : ( produc6174133586879617921nteger(code_divmod_integer(K,L)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(K),L) ) ).

% snd_divmod_integer
tff(fact_9024_mod__pure,axiom,
    ! [B: $o,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_o_assn(pure_assn,(B))),H2)
    <=> ( ( produc8586169260539613262et_nat(H2) = bot_bot_set_nat )
        & (B) ) ) ).

% mod_pure
tff(fact_9025_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : ( produc6174133586879617921nteger(code_divmod_abs(K,L)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)) ) ).

% snd_divmod_abs
tff(fact_9026_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_9027_mod__emp,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),H2)
    <=> ( produc8586169260539613262et_nat(H2) = bot_bot_set_nat ) ) ).

% mod_emp
tff(fact_9028_mod__star__trueE_H,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),top_top_assn)),H2)
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ( ( produc1824681642469235216et_nat(H5) = produc1824681642469235216et_nat(H2) )
           => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,produc8586169260539613262et_nat(H5)),produc8586169260539613262et_nat(H2))
             => ~ aa_Pro653082453696054616_nat_o(rep_assn(P2),H5) ) ) ) ).

% mod_star_trueE'
tff(fact_9029_one__mod__minus__numeral,axiom,
    ! [Na: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Na)),product_snd_int_int(unique5052692396658037445od_int(one,Na)))) ) ).

% one_mod_minus_numeral
tff(fact_9030_minus__one__mod__numeral,axiom,
    ! [Na: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Na)),product_snd_int_int(unique5052692396658037445od_int(one,Na))) ) ).

% minus_one_mod_numeral
tff(fact_9031_numeral__mod__minus__numeral,axiom,
    ! [M: num,Na: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Na)),product_snd_int_int(unique5052692396658037445od_int(M,Na)))) ) ).

% numeral_mod_minus_numeral
tff(fact_9032_minus__numeral__mod__numeral,axiom,
    ! [M: num,Na: num] : ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Na)),product_snd_int_int(unique5052692396658037445od_int(M,Na))) ) ).

% minus_numeral_mod_numeral
tff(fact_9033_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R: int] :
      ( aa_int_int(adjust_mod(L),R) = $ite(R = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R)) ) ).

% Divides.adjust_mod_def
tff(fact_9034_normalize__def,axiom,
    ! [P3: product_prod_int_int] :
      ( normalize(P3) = $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(P3)),
          $let(
            a2: int,
            a2:= aa_int_int(gcd_gcd_int(product_fst_int_int(P3)),product_snd_int_int(P3)),
            product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),a2),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),a2)) ),
          $ite(
            product_snd_int_int(P3) = zero_zero_int,
            product_Pair_int_int(zero_zero_int,one_one_int),
            $let(
              a2: int,
              a2:= aa_int_int(uminus_uminus_int,aa_int_int(gcd_gcd_int(product_fst_int_int(P3)),product_snd_int_int(P3))),
              product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),a2),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),a2)) ) ) ) ) ).

% normalize_def
tff(fact_9035_gcd__pos__int,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(gcd_gcd_int(M),Na))
    <=> ( ( M != zero_zero_int )
        | ( Na != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9036_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(gcd_gcd_int(A),B)),A) ) ).

% gcd_le1_int
tff(fact_9037_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(gcd_gcd_int(A),B)),B) ) ).

% gcd_le2_int
tff(fact_9038_gcd__cases__int,axiom,
    ! [Xa: int,Y: int,P2: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P2,aa_int_int(gcd_gcd_int(Xa),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xa)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P2,aa_int_int(gcd_gcd_int(Xa),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P2,aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,Xa)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(P2,aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,Xa)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P2,aa_int_int(gcd_gcd_int(Xa),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9039_gcd__ge__0__int,axiom,
    ! [Xa: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(gcd_gcd_int(Xa),Y)) ).

% gcd_ge_0_int
tff(fact_9040_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(gcd_gcd_int(A),B) ) ) ).

% gcd_unique_int
tff(fact_9041_gcd__non__0__int,axiom,
    ! [Y: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y)
     => ( aa_int_int(gcd_gcd_int(Xa),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9042_integer__set__bit__code,axiom,
    ! [Xa: code_integer,Na: code_integer,B: $o] :
      ( bits_integer_set_bit(Xa,Na,(B)) = $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger),
          aa_o_Code_integer(aa_Cod4043041029439962285nteger(aa_Cod3196565106923072233nteger(undefi7543901990407023336nteger,Xa),Na),(B)),
          $ite((B),aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(Xa),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_Code_integer_nat(code_nat_of_integer,Na)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(Xa),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_Code_integer_nat(code_nat_of_integer,Na)),one_one_Code_integer)))) ) ) ).

% integer_set_bit_code
tff(fact_9043_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_9044_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A) = A ) ).

% gcd_nat.left_neutral
tff(fact_9045_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_9046_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),zero_zero_nat) = A ) ).

% gcd_nat.right_neutral
tff(fact_9047_gcd__0__nat,axiom,
    ! [Xa: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),zero_zero_nat) = Xa ) ).

% gcd_0_nat
tff(fact_9048_gcd__0__left__nat,axiom,
    ! [Xa: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),Xa) = Xa ) ).

% gcd_0_left_nat
tff(fact_9049_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_9050_gcd__pos__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na))
    <=> ( ( M != zero_zero_nat )
        | ( Na != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9051_Gcd__in,axiom,
    ! [A4: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_set_nat_o(member_nat(A3),A4)
         => ( aa_set_nat_o(member_nat(B3),A4)
           => aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3)),A4) ) )
     => ( ( A4 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A4)),A4) ) ) ).

% Gcd_in
tff(fact_9052_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X: nat,Y3: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9053_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_9054_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).

% gcd_le1_nat
tff(fact_9055_gcd__diff1__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na) ) ) ).

% gcd_diff1_nat
tff(fact_9056_gcd__diff2__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na) ) ) ).

% gcd_diff2_nat
tff(fact_9057_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X: nat,Y3: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ) ).

% bezout_nat
tff(fact_9058_gcd__nat_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_9059_gcd__nat_Osimps,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y) = $ite(Y = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y))) ) ).

% gcd_nat.simps
tff(fact_9060_gcd__non__0__nat,axiom,
    ! [Y: nat,Xa: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9061_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_xm) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9062_gcd__is__Max__divisors__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aag(Na),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9063_gcd__nat_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,Xa,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xa),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9064_uint32__shiftl__def,axiom,
    ! [Xa: uint32,Na: code_integer] :
      ( aa_Cod8401701657498852871uint32(uint32_shiftl(Xa),Na) = $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Na) ),
          aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se5742574853984576102uint32),Xa),Na),
          aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,aa_Code_integer_nat(code_nat_of_integer,Na)),Xa) ) ) ).

% uint32_shiftl_def
tff(fact_9065_uint32__shiftr__def,axiom,
    ! [Xa: uint32,Na: code_integer] :
      ( aa_Cod8401701657498852871uint32(uint32_shiftr(Xa),Na) = $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Na) ),
          aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se3964402333458159761uint32),Xa),Na),
          aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,aa_Code_integer_nat(code_nat_of_integer,Na)),Xa) ) ) ).

% uint32_shiftr_def
tff(fact_9066_uint32__test__bit__def,axiom,
    ! [Xa: uint32,Na: code_integer] :
      ( aa_Code_integer_o(uint32_test_bit(Xa),Na)
    <=> $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),Na) ),
          aa_Code_integer_o(aa_uin5206268876086501546eger_o(aa_fun5262225984791465967eger_o(undefi9030187962712549518eger_o,bit_se5367290876889521763uint32),Xa),Na),
          aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xa),aa_Code_integer_nat(code_nat_of_integer,Na)) ) ) ).

% uint32_test_bit_def
tff(fact_9067_integer__set__bit__def,axiom,
    ! [Xa: code_integer,Na: code_integer,B: $o] :
      ( bits_integer_set_bit(Xa,Na,(B)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger),aa_o_Code_integer(aa_Cod4043041029439962285nteger(aa_Cod3196565106923072233nteger(undefi7543901990407023336nteger,Xa),Na),(B)),generi2397576812484419408nteger(Xa,aa_Code_integer_nat(code_nat_of_integer,Na),(B))) ) ).

% integer_set_bit_def
tff(fact_9068_uint32__shiftr__code,axiom,
    ! [W2: uint32,Na: code_integer] :
      ( rep_uint32(aa_Cod8401701657498852871uint32(uint32_shiftr(W2),Na)) = $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Na) ),
          rep_uint32(aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se3964402333458159761uint32),W2),Na)),
          bit_se5176125413884933531l_num1(aa_Code_integer_nat(code_nat_of_integer,Na),rep_uint32(W2)) ) ) ).

% uint32_shiftr_code
tff(fact_9069_uint32_Oless__iff__word__of,axiom,
    ! [P3: uint32,Q3: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_uint32,P3),Q3)
    <=> aa_wor4452366487969921764num1_o(aa_wor1562334678541590983num1_o(ord_le750835935415966154l_num1,rep_uint32(P3)),rep_uint32(Q3)) ) ).

% uint32.less_iff_word_of
tff(fact_9070_less__uint32_Orep__eq,axiom,
    ! [Xa: uint32,Xaa: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_uint32,Xa),Xaa)
    <=> aa_wor4452366487969921764num1_o(aa_wor1562334678541590983num1_o(ord_le750835935415966154l_num1,rep_uint32(Xa)),rep_uint32(Xaa)) ) ).

% less_uint32.rep_eq
tff(fact_9071_uint32_Oless__eq__iff__word__of,axiom,
    ! [P3: uint32,Q3: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_eq_uint32,P3),Q3)
    <=> aa_wor4452366487969921764num1_o(aa_wor1562334678541590983num1_o(ord_le3335648743751981014l_num1,rep_uint32(P3)),rep_uint32(Q3)) ) ).

% uint32.less_eq_iff_word_of
tff(fact_9072_less__eq__uint32_Orep__eq,axiom,
    ! [Xa: uint32,Xaa: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_eq_uint32,Xa),Xaa)
    <=> aa_wor4452366487969921764num1_o(aa_wor1562334678541590983num1_o(ord_le3335648743751981014l_num1,rep_uint32(Xa)),rep_uint32(Xaa)) ) ).

% less_eq_uint32.rep_eq
tff(fact_9073_uint32__test__bit__code,axiom,
    ! [W2: uint32,Na: code_integer] :
      ( aa_Code_integer_o(uint32_test_bit(W2),Na)
    <=> $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),Na) ),
          aa_Code_integer_o(aa_uin5206268876086501546eger_o(aa_fun5262225984791465967eger_o(undefi9030187962712549518eger_o,bit_se5367290876889521763uint32),W2),Na),
          aa_nat_o(bit_se6859397288646540909l_num1(rep_uint32(W2)),aa_Code_integer_nat(code_nat_of_integer,Na)) ) ) ).

% uint32_test_bit_code
tff(fact_9074_uint32__shiftl__code,axiom,
    ! [W2: uint32,Na: code_integer] :
      ( rep_uint32(aa_Cod8401701657498852871uint32(uint32_shiftl(W2),Na)) = $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Na) ),
          rep_uint32(aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se5742574853984576102uint32),W2),Na)),
          bit_se837345729053750000l_num1(aa_Code_integer_nat(code_nat_of_integer,Na),rep_uint32(W2)) ) ) ).

% uint32_shiftl_code
tff(fact_9075_integer__shiftl__def,axiom,
    ! [Xa: code_integer,Na: code_integer] :
      ( aa_Cod1783974652533396728nteger(bits_integer_shiftl(Xa),Na) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(undefi3911151285678701225nteger,Xa),Na),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_Code_integer_nat(code_nat_of_integer,Na)),Xa)) ) ).

% integer_shiftl_def
tff(fact_9076_less__eq__uint32_Orsp,axiom,
    bNF_re800061928031274251num1_o(fequal882237070059529185l_num1,bNF_re5013357767504289739m1_o_o(fequal882237070059529185l_num1,fequal_o),ord_le3335648743751981014l_num1,ord_le3335648743751981014l_num1) ).

% less_eq_uint32.rsp
tff(fact_9077_less__uint32_Orsp,axiom,
    bNF_re800061928031274251num1_o(fequal882237070059529185l_num1,bNF_re5013357767504289739m1_o_o(fequal882237070059529185l_num1,fequal_o),ord_le750835935415966154l_num1,ord_le750835935415966154l_num1) ).

% less_uint32.rsp
tff(fact_9078_uint32__set__bit__code,axiom,
    ! [W2: uint32,Na: code_integer,B: $o] :
      ( rep_uint32(aa_o_uint32(uint32_set_bit(W2,Na),(B))) = $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),Na) ),
          rep_uint32(aa_o_uint32(aa_Cod7056071456392079164uint32(aa_uin937759467983514409uint32(aa_fun8001877339868165167uint32(undefi8935238026407452494uint32,generi1993664874377053279uint32),W2),Na),(B))),
          generi5268133209446125161l_num1(rep_uint32(W2),aa_Code_integer_nat(code_nat_of_integer,Na),(B)) ) ) ).

% uint32_set_bit_code
tff(fact_9079_uint32__set__bit__def,axiom,
    ! [Xa: uint32,Na: code_integer,B: $o] :
      ( aa_o_uint32(uint32_set_bit(Xa,Na),(B)) = $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),Na) ),
          aa_o_uint32(aa_Cod7056071456392079164uint32(aa_uin937759467983514409uint32(aa_fun8001877339868165167uint32(undefi8935238026407452494uint32,generi1993664874377053279uint32),Xa),Na),(B)),
          aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,Xa),aa_Code_integer_nat(code_nat_of_integer,Na)),(B)) ) ) ).

% uint32_set_bit_def
tff(fact_9080_integer__shiftr__def,axiom,
    ! [Xa: code_integer,Na: code_integer] :
      ( aa_Cod1783974652533396728nteger(bits_integer_shiftr(Xa),Na) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(undefi3911151285678701225nteger,Xa),Na),aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(aa_Code_integer_nat(code_nat_of_integer,Na)),Xa)) ) ).

% integer_shiftr_def
tff(fact_9081_uint32__divmod__code,axiom,
    ! [Xa: uint32,Y: uint32] :
      ( uint32_divmod(Xa,Y) = $ite(
          aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_eq_uint32,numera9087168376688890119uint32(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,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,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,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))))))))))))))))))))))))))))))))),Y),
          $ite(aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_uint32,Xa),Y),produc1400373151660368625uint32(zero_zero_uint32,Xa),produc1400373151660368625uint32(one_one_uint32,minus_minus_uint32(Xa,Y))),
          $ite(
            Y = zero_zero_uint32,
            produc1400373151660368625uint32(div0_uint32(Xa),mod0_uint32(Xa)),
            $let(
              q: uint32,
              q:= aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,one_one_nat),uint32_sdiv(aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,one_one_nat),Xa),Y)),
              $let(
                r2: uint32,
                r2:= minus_minus_uint32(Xa,aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,q),Y)),
                $ite(aa_uint32_o(aa_uin5924812838015166427nt32_o(ord_less_eq_uint32,Y),r2),produc1400373151660368625uint32(aa_uint32_uint32(plus_plus_uint32(q),one_one_uint32),minus_minus_uint32(r2,Y)),produc1400373151660368625uint32(q,r2)) ) ) ) ) ) ).

% uint32_divmod_code
tff(fact_9082_uint32_Oset__bits__aux__code,axiom,
    ! [F: fun_nat_o,Na: nat,W2: uint32] :
      ( set_bits_aux_uint32(F,Na,W2) = $ite(
          Na = zero_zero_nat,
          W2,
          $let(
            n3: nat,
            n3:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),
            set_bits_aux_uint32(F,n3,
              bit_se2966626333419230250uint32(aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,one_one_nat),W2),
                $ite(aa_nat_o(F,n3),one_one_uint32,zero_zero_uint32))) ) ) ) ).

% uint32.set_bits_aux_code
tff(fact_9083_shiftr__uint32__code,axiom,
    ! [Na: nat,Xa: uint32] :
      ( aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,Na),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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,one))))))),aa_Cod8401701657498852871uint32(uint32_shiftr(Xa),aa_nat_Code_integer(code_integer_of_nat,Na)),zero_zero_uint32) ) ).

% shiftr_uint32_code
tff(fact_9084_nat__of__integer__integer__of__nat,axiom,
    ! [Na: nat] : ( aa_Code_integer_nat(code_nat_of_integer,aa_nat_Code_integer(code_integer_of_nat,Na)) = Na ) ).

% nat_of_integer_integer_of_nat
tff(fact_9085_integer__of__nat_Orep__eq,axiom,
    ! [Xa: nat] : ( aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,Xa)) = aa_nat_int(semiri1314217659103216013at_int,Xa) ) ).

% integer_of_nat.rep_eq
tff(fact_9086_int__of__integer__integer__of__nat,axiom,
    ! [Na: nat] : ( aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ) ).

% int_of_integer_integer_of_nat
tff(fact_9087_integer__of__nat__0,axiom,
    aa_nat_Code_integer(code_integer_of_nat,zero_zero_nat) = zero_z3403309356797280102nteger ).

% integer_of_nat_0
tff(fact_9088_integer__of__nat__1,axiom,
    aa_nat_Code_integer(code_integer_of_nat,one_one_nat) = one_one_Code_integer ).

% integer_of_nat_1
tff(fact_9089_integer__of__nat__less__0__conv,axiom,
    ! [Na: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(code_integer_of_nat,Na)),zero_z3403309356797280102nteger) ).

% integer_of_nat_less_0_conv
tff(fact_9090_integer__of__nat__eq__of__nat,axiom,
    code_integer_of_nat = semiri4939895301339042750nteger ).

% integer_of_nat_eq_of_nat
tff(fact_9091_integer__of__nat_Oabs__eq,axiom,
    ! [Xa: nat] : ( aa_nat_Code_integer(code_integer_of_nat,Xa) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% integer_of_nat.abs_eq
tff(fact_9092_integer__of__nat__numeral,axiom,
    ! [Na: num] : ( aa_nat_Code_integer(code_integer_of_nat,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ) ).

% integer_of_nat_numeral
tff(fact_9093_test__bit__uint32__code,axiom,
    ! [Xa: uint32,Na: nat] :
      ( aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xa),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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,one)))))))
        & aa_Code_integer_o(uint32_test_bit(Xa),aa_nat_Code_integer(code_integer_of_nat,Na)) ) ) ).

% test_bit_uint32_code
tff(fact_9094_set__bit__uint32__code,axiom,
    ! [Xa: uint32,Na: nat,B: $o] :
      ( aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,Xa),Na),(B)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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,one))))))),aa_o_uint32(uint32_set_bit(Xa,aa_nat_Code_integer(code_integer_of_nat,Na)),(B)),Xa) ) ).

% set_bit_uint32_code
tff(fact_9095_shiftl__uint32__code,axiom,
    ! [Na: nat,Xa: uint32] :
      ( aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,Na),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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,one))))))),aa_Cod8401701657498852871uint32(uint32_shiftl(Xa),aa_nat_Code_integer(code_integer_of_nat,Na)),zero_zero_uint32) ) ).

% shiftl_uint32_code
tff(fact_9096_sshiftr__uint32__code,axiom,
    ! [Na: nat,Xa: uint32] :
      ( aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,Na),Xa) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),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,one))))))),
          aa_Cod8401701657498852871uint32(uint32_sshiftr(Xa),aa_nat_Code_integer(code_integer_of_nat,Na)),
          $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),uminus_uminus_uint32(one_one_uint32),zero_zero_uint32) ) ) ).

% sshiftr_uint32_code
tff(fact_9097_uint32__sshiftr__def,axiom,
    ! [Xa: uint32,Na: code_integer] :
      ( aa_Cod8401701657498852871uint32(uint32_sshiftr(Xa),Na) = $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Na) ),
          aa_uint32_uint32(aa_Cod3565511682404152816uint32(aa_fun7349710212561521167uint32(undefi424500106705823974uint32,signed489701013188660438uint32),Na),Xa),
          aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,aa_Code_integer_nat(code_nat_of_integer,Na)),Xa) ) ) ).

% uint32_sshiftr_def
tff(fact_9098_uint32__sshiftr__code,axiom,
    ! [W2: uint32,Na: code_integer] :
      ( rep_uint32(aa_Cod8401701657498852871uint32(uint32_sshiftr(W2),Na)) = $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Na),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),Na) ),
          rep_uint32(aa_uint32_uint32(aa_Cod3565511682404152816uint32(aa_fun7349710212561521167uint32(undefi424500106705823974uint32,signed489701013188660438uint32),Na),W2)),
          signed5000768011106662067l_num1(aa_Code_integer_nat(code_nat_of_integer,Na),rep_uint32(W2)) ) ) ).

% uint32_sshiftr_code
tff(fact_9099_ATP_Olambda__1,axiom,
    ! [Uu2: nat] : ( aa_nat_real(aTP_Lamm_yd,Uu2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu2)) ) ).

% ATP.lambda_1
tff(fact_9100_ATP_Olambda__2,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_xh,Uu2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu2) = zero_zero_real ) ) ) ).

% ATP.lambda_2
tff(fact_9101_ATP_Olambda__3,axiom,
    ! [Uu2: nat] : ( aa_nat_real(aTP_Lamm_wl,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu2)) ) ).

% ATP.lambda_3
tff(fact_9102_ATP_Olambda__4,axiom,
    ! [Uu2: nat] : ( aa_nat_nat(aTP_Lamm_yi,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(suc,zero_zero_nat)) ) ).

% ATP.lambda_4
tff(fact_9103_ATP_Olambda__5,axiom,
    ! [Uu2: int] : ( aa_int_int(aTP_Lamm_aae,Uu2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),Uu2) ) ).

% ATP.lambda_5
tff(fact_9104_ATP_Olambda__6,axiom,
    ! [Uu2: real] : ( aa_real_real(aTP_Lamm_yu,Uu2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_xo,Uu2)) ) ).

% ATP.lambda_6
tff(fact_9105_ATP_Olambda__7,axiom,
    ! [Uu2: nat] : ( aa_nat_set_nat(aTP_Lamm_xn,Uu2) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xl,Uu2)) ) ).

% ATP.lambda_7
tff(fact_9106_ATP_Olambda__8,axiom,
    ! [Uu2: nat] : ( aa_nat_real(aTP_Lamm_zg,Uu2) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu2))) ) ).

% ATP.lambda_8
tff(fact_9107_ATP_Olambda__9,axiom,
    ! [Uu2: nat] : ( aa_nat8803788312118491388at_nat(aTP_Lamm_xs,Uu2) = product_Pair_nat_nat(aa_nat_nat(suc,Uu2)) ) ).

% ATP.lambda_9
tff(fact_9108_ATP_Olambda__10,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xk,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uu2),one_one_real)),aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_10
tff(fact_9109_ATP_Olambda__11,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_yb,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_11
tff(fact_9110_ATP_Olambda__12,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ya,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_12
tff(fact_9111_ATP_Olambda__13,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      ( aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_zb,Uu2),Uua) = $let(
          l2: int,
          l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu2)),
          $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ) ).

% ATP.lambda_13
tff(fact_9112_ATP_Olambda__14,axiom,
    ! [Uu2: real,Uua: int] :
      ( aa_int_o(aa_real_fun_int_o(aTP_Lamm_wt,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu2)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_14
tff(fact_9113_ATP_Olambda__15,axiom,
    ! [Uu2: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_wv(Uu2),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu2),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_15
tff(fact_9114_ATP_Olambda__16,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aTP_Lamm_uj(Uu2),Uua)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uua)),Uu2)
        & aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Uua) ) ) ).

% ATP.lambda_16
tff(fact_9115_ATP_Olambda__17,axiom,
    ! [Uu2: set_Code_integer,Uua: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aTP_Lamm_uo(Uu2),Uua)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uua)),Uu2)
        & distin1543349897113766820nteger(Uua) ) ) ).

% ATP.lambda_17
tff(fact_9116_ATP_Olambda__18,axiom,
    ! [Uu2: set_complex,Uua: list_complex] :
      ( aa_list_complex_o(aTP_Lamm_un(Uu2),Uua)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uua)),Uu2)
        & distinct_complex(Uua) ) ) ).

% ATP.lambda_18
tff(fact_9117_ATP_Olambda__19,axiom,
    ! [Uu2: set_real,Uua: list_real] :
      ( aa_list_real_o(aTP_Lamm_uk(Uu2),Uua)
    <=> ( aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Uua)),Uu2)
        & aa_list_real_o(distinct_real,Uua) ) ) ).

% ATP.lambda_19
tff(fact_9118_ATP_Olambda__20,axiom,
    ! [Uu2: set_o,Uua: list_o] :
      ( aa_list_o_o(aTP_Lamm_ul(Uu2),Uua)
    <=> ( aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Uua)),Uu2)
        & aa_list_o_o(distinct_o,Uua) ) ) ).

% ATP.lambda_20
tff(fact_9119_ATP_Olambda__21,axiom,
    ! [Uu2: set_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_um(Uu2),Uua)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Uua)),Uu2)
        & aa_list_nat_o(distinct_nat,Uua) ) ) ).

% ATP.lambda_21
tff(fact_9120_ATP_Olambda__22,axiom,
    ! [Uu2: set_int,Uua: list_int] :
      ( aa_list_int_o(aTP_Lamm_up(Uu2),Uua)
    <=> ( aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Uua)),Uu2)
        & distinct_int(Uua) ) ) ).

% ATP.lambda_22
tff(fact_9121_ATP_Olambda__23,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aTP_Lamm_qc(Uu2),Uua)
    <=> ( ( aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uua) = Uu2 )
        & aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Uua) ) ) ).

% ATP.lambda_23
tff(fact_9122_ATP_Olambda__24,axiom,
    ! [Uu2: set_real,Uua: list_real] :
      ( aa_list_real_o(aTP_Lamm_qe(Uu2),Uua)
    <=> ( ( aa_lis3769085826154824284t_real(set_real2,Uua) = Uu2 )
        & aa_list_real_o(distinct_real,Uua) ) ) ).

% ATP.lambda_24
tff(fact_9123_ATP_Olambda__25,axiom,
    ! [Uu2: set_o,Uua: list_o] :
      ( aa_list_o_o(aTP_Lamm_qf(Uu2),Uua)
    <=> ( ( aa_list_o_set_o(set_o2,Uua) = Uu2 )
        & aa_list_o_o(distinct_o,Uua) ) ) ).

% ATP.lambda_25
tff(fact_9124_ATP_Olambda__26,axiom,
    ! [Uu2: set_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_qd(Uu2),Uua)
    <=> ( ( aa_list_nat_set_nat(set_nat2,Uua) = Uu2 )
        & aa_list_nat_o(distinct_nat,Uua) ) ) ).

% ATP.lambda_26
tff(fact_9125_ATP_Olambda__27,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xo,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ) ).

% ATP.lambda_27
tff(fact_9126_ATP_Olambda__28,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yv,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% ATP.lambda_28
tff(fact_9127_ATP_Olambda__29,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_zc(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu2,Uua)) ) ).

% ATP.lambda_29
tff(fact_9128_ATP_Olambda__30,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_xm,Uu2),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu2),Uua)
        & ( Uu2 != Uua ) ) ) ).

% ATP.lambda_30
tff(fact_9129_ATP_Olambda__31,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_zv(Uu2),Uua)
    <=> ( aa_set_int_o(member_int(Uua),Uu2)
        & ! [X5: int] :
            ( aa_set_int_o(member_int(X5),Uu2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X5),Uua) ) ) ) ).

% ATP.lambda_31
tff(fact_9130_ATP_Olambda__32,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wm,Uu2),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ) ).

% ATP.lambda_32
tff(fact_9131_ATP_Olambda__33,axiom,
    ! [Uu2: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aTP_Lamm_qa(Uu2),Uua)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(Uua),Uu2)
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Uua) ) ) ).

% ATP.lambda_33
tff(fact_9132_ATP_Olambda__34,axiom,
    ! [Uu2: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_pz(Uu2),Uua)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu2)
        & aa_set_complex_o(finite3207457112153483333omplex,Uua) ) ) ).

% ATP.lambda_34
tff(fact_9133_ATP_Olambda__35,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_py,Uu2),Uua)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu2)
        & aa_set_nat_o(finite_finite_nat,Uua) ) ) ).

% ATP.lambda_35
tff(fact_9134_ATP_Olambda__36,axiom,
    ! [Uu2: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_qb(Uu2),Uua)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Uua),Uu2)
        & aa_set_int_o(finite_finite_int,Uua) ) ) ).

% ATP.lambda_36
tff(fact_9135_ATP_Olambda__37,axiom,
    ! [Uu2: nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_wg(Uu2),Uua) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uua) ) ).

% ATP.lambda_37
tff(fact_9136_ATP_Olambda__38,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_vy,Uu2),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu2) = one_one_complex ) ) ).

% ATP.lambda_38
tff(fact_9137_ATP_Olambda__39,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xg,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ) ).

% ATP.lambda_39
tff(fact_9138_ATP_Olambda__40,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_yc,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_40
tff(fact_9139_ATP_Olambda__41,axiom,
    ! [Uu2: fun_nat_uint32,Uua: nat] : ( aa_nat_uint32(aTP_Lamm_ks(Uu2),Uua) = minus_minus_uint32(aa_nat_uint32(Uu2,aa_nat_nat(suc,Uua)),aa_nat_uint32(Uu2,Uua)) ) ).

% ATP.lambda_41
tff(fact_9140_ATP_Olambda__42,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_ku(Uu2),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,aa_nat_nat(suc,Uua))),aa_nat_real(Uu2,Uua)) ) ).

% ATP.lambda_42
tff(fact_9141_ATP_Olambda__43,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_kt(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,aa_nat_nat(suc,Uua))),aa_nat_int(Uu2,Uua)) ) ).

% ATP.lambda_43
tff(fact_9142_ATP_Olambda__44,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : ( aa_nat_complex(aTP_Lamm_dh(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ) ).

% ATP.lambda_44
tff(fact_9143_ATP_Olambda__45,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_di(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ) ).

% ATP.lambda_45
tff(fact_9144_ATP_Olambda__46,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_dg(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ) ).

% ATP.lambda_46
tff(fact_9145_ATP_Olambda__47,axiom,
    ! [Uu2: fun_nat_uint32,Uua: nat] : ( aa_nat_uint32(aTP_Lamm_kv(Uu2),Uua) = minus_minus_uint32(aa_nat_uint32(Uu2,Uua),aa_nat_uint32(Uu2,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_47
tff(fact_9146_ATP_Olambda__48,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_kx(Uu2),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uua)),aa_nat_real(Uu2,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_48
tff(fact_9147_ATP_Olambda__49,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_kw(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu2,Uua)),aa_nat_int(Uu2,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_49
tff(fact_9148_ATP_Olambda__50,axiom,
    ! [Uu2: fun_Co1554742298434871494uint32,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_sq(Uu2),Uua)
    <=> ( aa_Cod8401701657498852871uint32(Uu2,Uua) = zero_zero_uint32 ) ) ).

% ATP.lambda_50
tff(fact_9149_ATP_Olambda__51,axiom,
    ! [Uu2: fun_Co3960896333048504454r_real,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_st(Uu2),Uua)
    <=> ( aa_Code_integer_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_51
tff(fact_9150_ATP_Olambda__52,axiom,
    ! [Uu2: fun_Code_integer_rat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_sw(Uu2),Uua)
    <=> ( aa_Code_integer_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_52
tff(fact_9151_ATP_Olambda__53,axiom,
    ! [Uu2: fun_complex_uint32,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_sp(Uu2),Uua)
    <=> ( aa_complex_uint32(Uu2,Uua) = zero_zero_uint32 ) ) ).

% ATP.lambda_53
tff(fact_9152_ATP_Olambda__54,axiom,
    ! [Uu2: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ss(Uu2),Uua)
    <=> ( aa_complex_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_54
tff(fact_9153_ATP_Olambda__55,axiom,
    ! [Uu2: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_sv(Uu2),Uua)
    <=> ( aa_complex_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_55
tff(fact_9154_ATP_Olambda__56,axiom,
    ! [Uu2: fun_int_uint32,Uua: int] :
      ( aa_int_o(aTP_Lamm_so(Uu2),Uua)
    <=> ( aa_int_uint32(Uu2,Uua) = zero_zero_uint32 ) ) ).

% ATP.lambda_56
tff(fact_9155_ATP_Olambda__57,axiom,
    ! [Uu2: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_sr(Uu2),Uua)
    <=> ( aa_int_real(Uu2,Uua) = zero_zero_real ) ) ).

% ATP.lambda_57
tff(fact_9156_ATP_Olambda__58,axiom,
    ! [Uu2: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_su(Uu2),Uua)
    <=> ( aa_int_rat(Uu2,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_58
tff(fact_9157_ATP_Olambda__59,axiom,
    ! [Uu2: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_sx(Uu2),Uua)
    <=> ( aa_int_nat(Uu2,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_59
tff(fact_9158_ATP_Olambda__60,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_zo(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_zc(Uu2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ) ).

% ATP.lambda_60
tff(fact_9159_ATP_Olambda__61,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_zn(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_zc(Uu2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_61
tff(fact_9160_ATP_Olambda__62,axiom,
    ! [Uu2: nat,Uua: real] : ( aa_real_real(aa_nat_fun_real_real(aTP_Lamm_wy,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu2)) ) ).

% ATP.lambda_62
tff(fact_9161_ATP_Olambda__63,axiom,
    ! [Uu2: $o,Uua: nat] : ( aa_nat_nat(aa_o_fun_nat_nat(aTP_Lamm_zz,(Uu2)),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Uu2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% ATP.lambda_63
tff(fact_9162_ATP_Olambda__64,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ww,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ) ).

% ATP.lambda_64
tff(fact_9163_ATP_Olambda__65,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xf,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ) ).

% ATP.lambda_65
tff(fact_9164_ATP_Olambda__66,axiom,
    ! [Uu2: num,Uua: num] : ( aa_num_int(aa_num_fun_num_int(aTP_Lamm_aaf,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu2)),aa_num_int(numeral_numeral_int,Uua)) ) ).

% ATP.lambda_66
tff(fact_9165_ATP_Olambda__67,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      ( aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_zp,Uu2),Uua) = $let(
          l2: num,
          l2:= code_num_of_integer(Uu2),
          $let(
            l3: num,
            l3:= aa_num_num(plus_plus_num(l2),l2),
            $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(plus_plus_num(l3),one)) ) ) ) ).

% ATP.lambda_67
tff(fact_9166_ATP_Olambda__68,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      ( aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_zq,Uu2),Uua) = $let(
          l2: nat,
          l2:= aa_Code_integer_nat(code_nat_of_integer,Uu2),
          $let(
            l3: nat,
            l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
            $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ) ).

% ATP.lambda_68
tff(fact_9167_ATP_Olambda__69,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zm,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ) ).

% ATP.lambda_69
tff(fact_9168_ATP_Olambda__70,axiom,
    ! [Uu2: int,Uua: int] : ( aa_int_int(aa_int_fun_int_int(aTP_Lamm_yg,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ) ).

% ATP.lambda_70
tff(fact_9169_ATP_Olambda__71,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zl,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ) ).

% ATP.lambda_71
tff(fact_9170_ATP_Olambda__72,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zi,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ) ).

% ATP.lambda_72
tff(fact_9171_ATP_Olambda__73,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_xz(Uu2),Uua) = groups6591440286371151544t_real(Uu2,aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ) ).

% ATP.lambda_73
tff(fact_9172_ATP_Olambda__74,axiom,
    ! [Uu2: assn,Uua: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(aTP_Lamm_tg,Uu2),Uua)
    <=> ord_less_eq_assn(Uua,Uu2) ) ).

% ATP.lambda_74
tff(fact_9173_ATP_Olambda__75,axiom,
    ! [Uu2: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aTP_Lamm_ct(Uu2),Uua)
    <=> aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(Uua),Uu2) ) ).

% ATP.lambda_75
tff(fact_9174_ATP_Olambda__76,axiom,
    ! [Uu2: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_cs(Uu2),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu2) ) ).

% ATP.lambda_76
tff(fact_9175_ATP_Olambda__77,axiom,
    ! [Uu2: set_o,Uua: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_ti,Uu2),Uua)
    <=> aa_set_o_o(ord_less_eq_set_o(Uua),Uu2) ) ).

% ATP.lambda_77
tff(fact_9176_ATP_Olambda__78,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_cr,Uu2),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu2) ) ).

% ATP.lambda_78
tff(fact_9177_ATP_Olambda__79,axiom,
    ! [Uu2: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_cu,Uu2),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu2) ) ).

% ATP.lambda_79
tff(fact_9178_ATP_Olambda__80,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bs,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu2) ) ).

% ATP.lambda_80
tff(fact_9179_ATP_Olambda__81,axiom,
    ! [Uu2: nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_wx(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu2) ) ).

% ATP.lambda_81
tff(fact_9180_ATP_Olambda__82,axiom,
    ! [Uu2: assn,Uua: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(aTP_Lamm_th,Uu2),Uua)
    <=> ord_less_assn(Uua,Uu2) ) ).

% ATP.lambda_82
tff(fact_9181_ATP_Olambda__83,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ff,Uu2),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu2) ) ).

% ATP.lambda_83
tff(fact_9182_ATP_Olambda__84,axiom,
    ! [Uu2: set_o,Uua: set_o] :
      ( aa_set_o_o(aa_set_o_fun_set_o_o(aTP_Lamm_tj,Uu2),Uua)
    <=> aa_set_o_o(ord_less_set_o(Uua),Uu2) ) ).

% ATP.lambda_84
tff(fact_9183_ATP_Olambda__85,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_fb,Uu2),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uua),Uu2) ) ).

% ATP.lambda_85
tff(fact_9184_ATP_Olambda__86,axiom,
    ! [Uu2: set_int,Uua: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_tk,Uu2),Uua)
    <=> aa_set_int_o(ord_less_set_int(Uua),Uu2) ) ).

% ATP.lambda_86
tff(fact_9185_ATP_Olambda__87,axiom,
    ! [Uu2: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_fc,Uu2),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu2) ) ).

% ATP.lambda_87
tff(fact_9186_ATP_Olambda__88,axiom,
    ! [Uu2: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_fd,Uu2),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu2) ) ).

% ATP.lambda_88
tff(fact_9187_ATP_Olambda__89,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_br,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu2) ) ).

% ATP.lambda_89
tff(fact_9188_ATP_Olambda__90,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fe,Uu2),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu2) ) ).

% ATP.lambda_90
tff(fact_9189_ATP_Olambda__91,axiom,
    ! [Uu2: nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_zd(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu2) ) ).

% ATP.lambda_91
tff(fact_9190_ATP_Olambda__92,axiom,
    ! [Uu2: nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_vk(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu2) ) ).

% ATP.lambda_92
tff(fact_9191_ATP_Olambda__93,axiom,
    ! [Uu2: nat,Uua: real] : ( aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yj,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) ) ).

% ATP.lambda_93
tff(fact_9192_ATP_Olambda__94,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : ( aa_Cod1783974652533396728nteger(aTP_Lamm_xe(Uu2),Uua) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uua),Uu2) ) ).

% ATP.lambda_94
tff(fact_9193_ATP_Olambda__95,axiom,
    ! [Uu2: nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_yh(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu2) ) ).

% ATP.lambda_95
tff(fact_9194_ATP_Olambda__96,axiom,
    ! [Uu2: real,Uua: real] : ( aa_real_real(aTP_Lamm_yk(Uu2),Uua) = aa_real_real(powr_real(Uua),Uu2) ) ).

% ATP.lambda_96
tff(fact_9195_ATP_Olambda__97,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_xl,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu2) ) ).

% ATP.lambda_97
tff(fact_9196_ATP_Olambda__98,axiom,
    ! [Uu2: nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_wf(Uu2),Uua) = aa_nat_nat(binomial(Uua),Uu2) ) ).

% ATP.lambda_98
tff(fact_9197_ATP_Olambda__99,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zh,Uu2),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu2) ) ).

% ATP.lambda_99
tff(fact_9198_ATP_Olambda__100,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_a(Uu2),Uua)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uua),Uu2) ) ).

% ATP.lambda_100
tff(fact_9199_ATP_Olambda__101,axiom,
    ! [Uu2: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ab(Uu2),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu2) ) ).

% ATP.lambda_101
tff(fact_9200_ATP_Olambda__102,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aa(Uu2),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu2) ) ).

% ATP.lambda_102
tff(fact_9201_ATP_Olambda__103,axiom,
    ! [Uu2: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ac(Uu2),Uua)
    <=> member_list_nat(Uua,Uu2) ) ).

% ATP.lambda_103
tff(fact_9202_ATP_Olambda__104,axiom,
    ! [Uu2: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ad(Uu2),Uua)
    <=> aa_set_set_nat_o(member_set_nat(Uua),Uu2) ) ).

% ATP.lambda_104
tff(fact_9203_ATP_Olambda__105,axiom,
    ! [Uu2: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ae(Uu2),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu2) ) ).

% ATP.lambda_105
tff(fact_9204_ATP_Olambda__106,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_af(Uu2),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu2) ) ).

% ATP.lambda_106
tff(fact_9205_ATP_Olambda__107,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(aTP_Lamm_bk,Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_107
tff(fact_9206_ATP_Olambda__108,axiom,
    ! [Uu2: complex,Uua: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_bl,Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_108
tff(fact_9207_ATP_Olambda__109,axiom,
    ! [Uu2: list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_bm(Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_109
tff(fact_9208_ATP_Olambda__110,axiom,
    ! [Uu2: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_bq,(Uu2)),(Uua))
    <=> ( (Uua)
      <=> (Uu2) ) ) ).

% ATP.lambda_110
tff(fact_9209_ATP_Olambda__111,axiom,
    ! [Uu2: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_bn,Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_111
tff(fact_9210_ATP_Olambda__112,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bo,Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_112
tff(fact_9211_ATP_Olambda__113,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bp,Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_113
tff(fact_9212_ATP_Olambda__114,axiom,
    ! [Uu2: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_mw(Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Uu2,Uua)) ) ).

% ATP.lambda_114
tff(fact_9213_ATP_Olambda__115,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_mu(Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(Uu2,Uua)) ) ).

% ATP.lambda_115
tff(fact_9214_ATP_Olambda__116,axiom,
    ! [Uu2: fun_list_nat_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_mm(Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_list_nat_nat(Uu2,Uua)) ) ).

% ATP.lambda_116
tff(fact_9215_ATP_Olambda__117,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_mo(Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(Uu2,Uua)) ) ).

% ATP.lambda_117
tff(fact_9216_ATP_Olambda__118,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_mq(Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu2,Uua)) ) ).

% ATP.lambda_118
tff(fact_9217_ATP_Olambda__119,axiom,
    ! [Uu2: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ms(Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Uu2,Uua)) ) ).

% ATP.lambda_119
tff(fact_9218_ATP_Olambda__120,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_kr(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_120
tff(fact_9219_ATP_Olambda__121,axiom,
    ! [Uu2: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_wn(Uu2),Uua)
    <=> aa_nat_o(Uu2,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_121
tff(fact_9220_ATP_Olambda__122,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : ( aa_nat_rat(aTP_Lamm_ko(Uu2),Uua) = aa_nat_rat(Uu2,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_122
tff(fact_9221_ATP_Olambda__123,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_kq(Uu2),Uua) = aa_nat_nat(Uu2,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_123
tff(fact_9222_ATP_Olambda__124,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : ( aa_nat_int(aTP_Lamm_kp(Uu2),Uua) = aa_nat_int(Uu2,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_124
tff(fact_9223_ATP_Olambda__125,axiom,
    ! [Uu2: fun_nat_real,Uua: real] : ( aa_real_real(aTP_Lamm_yy(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_yx(Uu2),Uua)) ) ).

% ATP.lambda_125
tff(fact_9224_ATP_Olambda__126,axiom,
    ! [Uu2: real,Uua: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zk,Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ) ).

% ATP.lambda_126
tff(fact_9225_ATP_Olambda__127,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_ep(Uu2),Uua)
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uua),Uu2) ) ).

% ATP.lambda_127
tff(fact_9226_ATP_Olambda__128,axiom,
    ! [Uu2: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_er(Uu2),Uua)
    <=> ~ aa_set_complex_o(member_complex(Uua),Uu2) ) ).

% ATP.lambda_128
tff(fact_9227_ATP_Olambda__129,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_eq(Uu2),Uua)
    <=> ~ aa_set_real_o(member_real(Uua),Uu2) ) ).

% ATP.lambda_129
tff(fact_9228_ATP_Olambda__130,axiom,
    ! [Uu2: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_es(Uu2),Uua)
    <=> ~ member_list_nat(Uua,Uu2) ) ).

% ATP.lambda_130
tff(fact_9229_ATP_Olambda__131,axiom,
    ! [Uu2: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_et(Uu2),Uua)
    <=> ~ aa_set_set_nat_o(member_set_nat(Uua),Uu2) ) ).

% ATP.lambda_131
tff(fact_9230_ATP_Olambda__132,axiom,
    ! [Uu2: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_eu(Uu2),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu2) ) ).

% ATP.lambda_132
tff(fact_9231_ATP_Olambda__133,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ev(Uu2),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu2) ) ).

% ATP.lambda_133
tff(fact_9232_ATP_Olambda__134,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_zf(Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu2,Uua)) ) ).

% ATP.lambda_134
tff(fact_9233_ATP_Olambda__135,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : ( aa_nat_real(aTP_Lamm_xx(Uu2),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu2,Uua)) ) ).

% ATP.lambda_135
tff(fact_9234_ATP_Olambda__136,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: real] : ( aa_real_real(aTP_Lamm_za(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(Uu2,Uua)) ) ).

% ATP.lambda_136
tff(fact_9235_ATP_Olambda__137,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex] : ( aa_complex_nat(aTP_Lamm_sy(Uu2),Uua) = aa_nat_nat(suc,aa_complex_nat(Uu2,Uua)) ) ).

% ATP.lambda_137
tff(fact_9236_ATP_Olambda__138,axiom,
    ! [Uu2: fun_literal_nat,Uua: literal] : ( aa_literal_nat(aTP_Lamm_sz(Uu2),Uua) = aa_nat_nat(suc,aa_literal_nat(Uu2,Uua)) ) ).

% ATP.lambda_138
tff(fact_9237_ATP_Olambda__139,axiom,
    ! [Uu2: fun_list_nat_nat,Uua: list_nat] : ( aa_list_nat_nat(aTP_Lamm_ta(Uu2),Uua) = aa_nat_nat(suc,aa_list_nat_nat(Uu2,Uua)) ) ).

% ATP.lambda_139
tff(fact_9238_ATP_Olambda__140,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: set_nat] : ( aa_set_nat_nat(aTP_Lamm_tb(Uu2),Uua) = aa_nat_nat(suc,aa_set_nat_nat(Uu2,Uua)) ) ).

% ATP.lambda_140
tff(fact_9239_ATP_Olambda__141,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_tc(Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(Uu2,Uua)) ) ).

% ATP.lambda_141
tff(fact_9240_ATP_Olambda__142,axiom,
    ! [Uu2: fun_complex_o,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ew(Uu2),Uua)
    <=> ~ aa_complex_o(Uu2,Uua) ) ).

% ATP.lambda_142
tff(fact_9241_ATP_Olambda__143,axiom,
    ! [Uu2: fun_list_nat_o,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ex(Uu2),Uua)
    <=> ~ aa_list_nat_o(Uu2,Uua) ) ).

% ATP.lambda_143
tff(fact_9242_ATP_Olambda__144,axiom,
    ! [Uu2: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ey(Uu2),Uua)
    <=> ~ aa_set_nat_o(Uu2,Uua) ) ).

% ATP.lambda_144
tff(fact_9243_ATP_Olambda__145,axiom,
    ! [Uu2: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ez(Uu2),Uua)
    <=> ~ aa_nat_o(Uu2,Uua) ) ).

% ATP.lambda_145
tff(fact_9244_ATP_Olambda__146,axiom,
    ! [Uu2: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_fa(Uu2),Uua)
    <=> ~ aa_int_o(Uu2,Uua) ) ).

% ATP.lambda_146
tff(fact_9245_ATP_Olambda__147,axiom,
    ! [Uu2: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zw(Uu2),Uua)
    <=> ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),M5)
         => ( aa_nat_o(Uu2,M5)
          <=> aa_nat_o(Uu2,Uua) ) ) ) ).

% ATP.lambda_147
tff(fact_9246_ATP_Olambda__148,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_zu(Uu2),Uua)
    <=> ! [X5: real] :
          ( aa_set_real_o(member_real(X5),Uu2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X5),Uua) ) ) ).

% ATP.lambda_148
tff(fact_9247_ATP_Olambda__149,axiom,
    ! [Uu2: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zy(Uu2),Uua)
    <=> ! [N10: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),N10)
         => aa_nat_o(Uu2,N10) ) ) ).

% ATP.lambda_149
tff(fact_9248_ATP_Olambda__150,axiom,
    ! [Uu2: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zx(Uu2),Uua)
    <=> ! [N10: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),N10)
         => ~ aa_nat_o(Uu2,N10) ) ) ).

% ATP.lambda_150
tff(fact_9249_ATP_Olambda__151,axiom,
    ! [Uu2: nat,Uua: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_cn,Uu2),Uua) = aa_nat_nat(suc,Uu2) ) ).

% ATP.lambda_151
tff(fact_9250_ATP_Olambda__152,axiom,
    ! [Uu2: nat,Uua: real] : ( aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_co,Uu2),Uua) = aa_nat_nat(suc,Uu2) ) ).

% ATP.lambda_152
tff(fact_9251_ATP_Olambda__153,axiom,
    ! [Uu2: nat,Uua: $o] : ( aa_o_nat(aa_nat_fun_o_nat(aTP_Lamm_cp,Uu2),(Uua)) = aa_nat_nat(suc,Uu2) ) ).

% ATP.lambda_153
tff(fact_9252_ATP_Olambda__154,axiom,
    ! [Uu2: nat,Uua: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cq,Uu2),Uua) = aa_nat_nat(suc,Uu2) ) ).

% ATP.lambda_154
tff(fact_9253_ATP_Olambda__155,axiom,
    ! [Uu2: num,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_xt(Uu2),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu2)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uub),aa_num_Code_integer(numera6620942414471956472nteger,Uu2))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ) ).

% ATP.lambda_155
tff(fact_9254_ATP_Olambda__156,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] :
      ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_xu(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu2)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu2))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ) ).

% ATP.lambda_156
tff(fact_9255_ATP_Olambda__157,axiom,
    ! [Uu2: num,Uua: int,Uub: int] :
      ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_xv(Uu2),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu2)),Uub),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu2))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ) ).

% ATP.lambda_157
tff(fact_9256_ATP_Olambda__158,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_uint32(aa_fun7720134271958203516uint32(aTP_Lamm_bt(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_VEBT_VEBT_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_158
tff(fact_9257_ATP_Olambda__159,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_bz(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_VEBT_VEBT_real(Uua,Uub),zero_zero_real) ) ).

% ATP.lambda_159
tff(fact_9258_ATP_Olambda__160,axiom,
    ! [Uu2: code_integer,Uua: fun_Co1554742298434871494uint32,Uub: code_integer] :
      ( aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(aTP_Lamm_by(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_Cod8401701657498852871uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_160
tff(fact_9259_ATP_Olambda__161,axiom,
    ! [Uu2: complex,Uua: fun_complex_uint32,Uub: complex] :
      ( aa_complex_uint32(aa_fun947202747272686700uint32(aTP_Lamm_bx(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_complex_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_161
tff(fact_9260_ATP_Olambda__162,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_cc(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ) ).

% ATP.lambda_162
tff(fact_9261_ATP_Olambda__163,axiom,
    ! [Uu2: real,Uua: fun_real_uint32,Uub: real] :
      ( aa_real_uint32(aa_fun39000138016665704uint32(aTP_Lamm_bu(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_real_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_163
tff(fact_9262_ATP_Olambda__164,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: real] :
      ( aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ca(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_real_real(Uua,Uub),zero_zero_real) ) ).

% ATP.lambda_164
tff(fact_9263_ATP_Olambda__165,axiom,
    ! [Uu2: nat,Uua: fun_nat_uint32,Uub: nat] :
      ( aa_nat_uint32(aa_fun7322500112117231024uint32(aTP_Lamm_bv(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_nat_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_165
tff(fact_9264_ATP_Olambda__166,axiom,
    ! [Uu2: int,Uua: fun_int_uint32,Uub: int] :
      ( aa_int_uint32(aa_fun7636875859098149992uint32(aTP_Lamm_bw(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_int_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_166
tff(fact_9265_ATP_Olambda__167,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: int] :
      ( aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_cb(Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa_int_real(Uua,Uub),zero_zero_real) ) ).

% ATP.lambda_167
tff(fact_9266_ATP_Olambda__168,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_uint32(aa_fun7720134271958203516uint32(aTP_Lamm_cd(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_VEBT_VEBT_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_168
tff(fact_9267_ATP_Olambda__169,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_cj(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_VEBT_VEBT_real(Uua,Uub),zero_zero_real) ) ).

% ATP.lambda_169
tff(fact_9268_ATP_Olambda__170,axiom,
    ! [Uu2: code_integer,Uua: fun_Co1554742298434871494uint32,Uub: code_integer] :
      ( aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(aTP_Lamm_ci(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_Cod8401701657498852871uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_170
tff(fact_9269_ATP_Olambda__171,axiom,
    ! [Uu2: complex,Uua: fun_complex_uint32,Uub: complex] :
      ( aa_complex_uint32(aa_fun947202747272686700uint32(aTP_Lamm_ch(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_complex_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_171
tff(fact_9270_ATP_Olambda__172,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_cm(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_complex_real(Uua,Uub),zero_zero_real) ) ).

% ATP.lambda_172
tff(fact_9271_ATP_Olambda__173,axiom,
    ! [Uu2: real,Uua: fun_real_uint32,Uub: real] :
      ( aa_real_uint32(aa_fun39000138016665704uint32(aTP_Lamm_ce(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_real_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_173
tff(fact_9272_ATP_Olambda__174,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: real] :
      ( aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ck(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_real_real(Uua,Uub),zero_zero_real) ) ).

% ATP.lambda_174
tff(fact_9273_ATP_Olambda__175,axiom,
    ! [Uu2: nat,Uua: fun_nat_uint32,Uub: nat] :
      ( aa_nat_uint32(aa_fun7322500112117231024uint32(aTP_Lamm_cf(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_nat_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_175
tff(fact_9274_ATP_Olambda__176,axiom,
    ! [Uu2: int,Uua: fun_int_uint32,Uub: int] :
      ( aa_int_uint32(aa_fun7636875859098149992uint32(aTP_Lamm_cg(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_int_uint32(Uua,Uub),zero_zero_uint32) ) ).

% ATP.lambda_176
tff(fact_9275_ATP_Olambda__177,axiom,
    ! [Uu2: int,Uua: fun_int_real,Uub: int] :
      ( aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_cl(Uu2),Uua),Uub) = $ite(Uub = Uu2,aa_int_real(Uua,Uub),zero_zero_real) ) ).

% ATP.lambda_177
tff(fact_9276_ATP_Olambda__178,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aac(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu2)),Uub))) ) ).

% ATP.lambda_178
tff(fact_9277_ATP_Olambda__179,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aaa(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Uu2)),Uub))) ) ).

% ATP.lambda_179
tff(fact_9278_ATP_Olambda__180,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aab(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uu2),Uub))) ) ).

% ATP.lambda_180
tff(fact_9279_ATP_Olambda__181,axiom,
    ! [Uu2: fun_VEBT_VEBT_uint32,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_uint32(aa_fun2089568865546376974uint32(aTP_Lamm_my(Uu2),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_uint32(Uu2,Uub),zero_zero_uint32) ) ).

% ATP.lambda_181
tff(fact_9280_ATP_Olambda__182,axiom,
    ! [Uu2: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_real(aa_fun3597767889293385422T_real(aTP_Lamm_nf(Uu2),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_real(Uu2,Uub),zero_zero_real) ) ).

% ATP.lambda_182
tff(fact_9281_ATP_Olambda__183,axiom,
    ! [Uu2: fun_Co1554742298434871494uint32,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Cod8401701657498852871uint32(aa_fun7787524580160642142uint32(aTP_Lamm_ne(Uu2),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Cod8401701657498852871uint32(Uu2,Uub),zero_zero_uint32) ) ).

% ATP.lambda_183
tff(fact_9282_ATP_Olambda__184,axiom,
    ! [Uu2: fun_complex_uint32,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_uint32(aa_fun9209306604415416350uint32(aTP_Lamm_nc(Uu2),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_uint32(Uu2,Uub),zero_zero_uint32) ) ).

% ATP.lambda_184
tff(fact_9283_ATP_Olambda__185,axiom,
    ! [Uu2: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_ni(Uu2),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu2,Uub),zero_zero_real) ) ).

% ATP.lambda_185
tff(fact_9284_ATP_Olambda__186,axiom,
    ! [Uu2: fun_real_uint32,Uua: fun_real_o,Uub: real] :
      ( aa_real_uint32(aa_fun7747623534341301918uint32(aTP_Lamm_mz(Uu2),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_uint32(Uu2,Uub),zero_zero_uint32) ) ).

% ATP.lambda_186
tff(fact_9285_ATP_Olambda__187,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_ng(Uu2),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu2,Uub),zero_zero_real) ) ).

% ATP.lambda_187
tff(fact_9286_ATP_Olambda__188,axiom,
    ! [Uu2: fun_nat_uint32,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_uint32(aa_fun7328656920723993502uint32(aTP_Lamm_na(Uu2),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_uint32(Uu2,Uub),zero_zero_uint32) ) ).

% ATP.lambda_188
tff(fact_9287_ATP_Olambda__189,axiom,
    ! [Uu2: fun_int_uint32,Uua: fun_int_o,Uub: int] :
      ( aa_int_uint32(aa_fun1392314963150257310uint32(aTP_Lamm_nb(Uu2),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_uint32(Uu2,Uub),zero_zero_uint32) ) ).

% ATP.lambda_189
tff(fact_9288_ATP_Olambda__190,axiom,
    ! [Uu2: fun_int_real,Uua: fun_int_o,Uub: int] :
      ( aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_nh(Uu2),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu2,Uub),zero_zero_real) ) ).

% ATP.lambda_190
tff(fact_9289_ATP_Olambda__191,axiom,
    ! [Uu2: fun_VE2006399705683203104omplex,Uua: complex,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_complex(aa_com2862003112617944841omplex(aTP_Lamm_jj(Uu2),Uua),Uub) = aa_complex_complex(aa_VEB5608662635949521omplex(Uu2,Uub),Uua) ) ).

% ATP.lambda_191
tff(fact_9290_ATP_Olambda__192,axiom,
    ! [Uu2: fun_VE597728521808591204at_nat,Uua: nat,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_im(Uu2),Uua),Uub) = aa_nat_nat(aa_VEB1122092716774696213at_nat(Uu2,Uub),Uua) ) ).

% ATP.lambda_192
tff(fact_9291_ATP_Olambda__193,axiom,
    ! [Uu2: fun_Co8905303033605938227omplex,Uua: complex,Uub: code_integer] : ( aa_Cod5621350732091595593omplex(aa_com7552888802784664052omplex(aTP_Lamm_kb(Uu2),Uua),Uub) = aa_complex_complex(aa_Cod3397930797123873268omplex(Uu2,Uub),Uua) ) ).

% ATP.lambda_193
tff(fact_9292_ATP_Olambda__194,axiom,
    ! [Uu2: fun_Co3312552843764780407at_nat,Uua: nat,Uub: code_integer] : ( aa_Code_integer_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_je(Uu2),Uua),Uub) = aa_nat_nat(aa_Cod8670293052248272184at_nat(Uu2,Uub),Uua) ) ).

% ATP.lambda_194
tff(fact_9293_ATP_Olambda__195,axiom,
    ! [Uu2: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : ( aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_iz(Uu2),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu2,Uub),Uua) ) ).

% ATP.lambda_195
tff(fact_9294_ATP_Olambda__196,axiom,
    ! [Uu2: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : ( aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_jn(Uu2),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu2,Uub),Uua) ) ).

% ATP.lambda_196
tff(fact_9295_ATP_Olambda__197,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : ( aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yz(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu2,Uub),Uua) ) ).

% ATP.lambda_197
tff(fact_9296_ATP_Olambda__198,axiom,
    ! [Uu2: fun_real_fun_nat_nat,Uua: nat,Uub: real] : ( aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_iq(Uu2),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu2,Uub),Uua) ) ).

% ATP.lambda_198
tff(fact_9297_ATP_Olambda__199,axiom,
    ! [Uu2: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : ( aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jr(Uu2),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu2,Uub),Uua) ) ).

% ATP.lambda_199
tff(fact_9298_ATP_Olambda__200,axiom,
    ! [Uu2: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : ( aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_jw(Uu2),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu2,Uub),Uua) ) ).

% ATP.lambda_200
tff(fact_9299_ATP_Olambda__201,axiom,
    ! [Uu2: fun_int_fun_nat_nat,Uua: nat,Uub: int] : ( aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_iu(Uu2),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu2,Uub),Uua) ) ).

% ATP.lambda_201
tff(fact_9300_ATP_Olambda__202,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      ( aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_aad(Uu2),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu2),Uua,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),Uub = one_one_Code_integer) ) ).

% ATP.lambda_202
tff(fact_9301_ATP_Olambda__203,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yn(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ) ).

% ATP.lambda_203
tff(fact_9302_ATP_Olambda__204,axiom,
    ! [Uu2: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : ( aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_yo(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ) ).

% ATP.lambda_204
tff(fact_9303_ATP_Olambda__205,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yw(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ) ).

% ATP.lambda_205
tff(fact_9304_ATP_Olambda__206,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : ( aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_wk(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ) ).

% ATP.lambda_206
tff(fact_9305_ATP_Olambda__207,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_wr(Uu2),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),one_one_nat) )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_207
tff(fact_9306_ATP_Olambda__208,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ws(Uu2),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu2 )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_list_nat_nat(groups4561878855575611511st_nat,Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_208
tff(fact_9307_ATP_Olambda__209,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_dl(Uu2),Uua),Uub)
    <=> ( ( size_s6755466524823107622T_VEBT(Uub) = Uua )
        & aa_list_VEBT_VEBT_o(distinct_VEBT_VEBT,Uub)
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uub)),Uu2) ) ) ).

% ATP.lambda_209
tff(fact_9308_ATP_Olambda__210,axiom,
    ! [Uu2: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_dk(Uu2),Uua),Uub)
    <=> ( ( size_s3445333598471063425nteger(Uub) = Uua )
        & distin1543349897113766820nteger(Uub)
        & aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu2) ) ) ).

% ATP.lambda_210
tff(fact_9309_ATP_Olambda__211,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_dj(Uu2),Uua),Uub)
    <=> ( ( size_s3451745648224563538omplex(Uub) = Uua )
        & distinct_complex(Uub)
        & aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu2) ) ) ).

% ATP.lambda_211
tff(fact_9310_ATP_Olambda__212,axiom,
    ! [Uu2: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_dm(Uu2),Uua),Uub)
    <=> ( ( size_size_list_real(Uub) = Uua )
        & aa_list_real_o(distinct_real,Uub)
        & aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Uub)),Uu2) ) ) ).

% ATP.lambda_212
tff(fact_9311_ATP_Olambda__213,axiom,
    ! [Uu2: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_dn(Uu2),Uua),Uub)
    <=> ( ( size_size_list_o(Uub) = Uua )
        & aa_list_o_o(distinct_o,Uub)
        & aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Uub)),Uu2) ) ) ).

% ATP.lambda_213
tff(fact_9312_ATP_Olambda__214,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_do(Uu2),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua )
        & aa_list_nat_o(distinct_nat,Uub)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Uub)),Uu2) ) ) ).

% ATP.lambda_214
tff(fact_9313_ATP_Olambda__215,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_dp(Uu2),Uua),Uub)
    <=> ( ( size_size_list_int(Uub) = Uua )
        & distinct_int(Uub)
        & aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Uub)),Uu2) ) ) ).

% ATP.lambda_215
tff(fact_9314_ATP_Olambda__216,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_vf(Uu2),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_216
tff(fact_9315_ATP_Olambda__217,axiom,
    ! [Uu2: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_ve(Uu2),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3445333598471063425nteger(Uub)),Uua) ) ) ).

% ATP.lambda_217
tff(fact_9316_ATP_Olambda__218,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_vd(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_218
tff(fact_9317_ATP_Olambda__219,axiom,
    ! [Uu2: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_vg(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Uub)),Uua) ) ) ).

% ATP.lambda_219
tff(fact_9318_ATP_Olambda__220,axiom,
    ! [Uu2: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_vh(Uu2),Uua),Uub)
    <=> ( aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_220
tff(fact_9319_ATP_Olambda__221,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_vi(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_list_nat_nat(size_size_list_nat,Uub)),Uua) ) ) ).

% ATP.lambda_221
tff(fact_9320_ATP_Olambda__222,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_vj(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_222
tff(fact_9321_ATP_Olambda__223,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_ue(Uu2),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uub)),Uu2)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_223
tff(fact_9322_ATP_Olambda__224,axiom,
    ! [Uu2: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_ud(Uu2),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu2)
        & ( size_s3445333598471063425nteger(Uub) = Uua ) ) ) ).

% ATP.lambda_224
tff(fact_9323_ATP_Olambda__225,axiom,
    ! [Uu2: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_uc(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu2)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_225
tff(fact_9324_ATP_Olambda__226,axiom,
    ! [Uu2: set_literal,Uua: nat,Uub: list_literal] :
      ( aa_list_literal_o(aa_nat1159182957892526735eral_o(aTP_Lamm_va(Uu2),Uua),Uub)
    <=> ( ord_le7307670543136651348iteral(set_literal2(Uub),Uu2)
        & ( size_s2501651207091587910iteral(Uub) = Uua ) ) ) ).

% ATP.lambda_226
tff(fact_9325_ATP_Olambda__227,axiom,
    ! [Uu2: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_uf(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(aa_lis3769085826154824284t_real(set_real2,Uub)),Uu2)
        & ( size_size_list_real(Uub) = Uua ) ) ) ).

% ATP.lambda_227
tff(fact_9326_ATP_Olambda__228,axiom,
    ! [Uu2: set_list_nat,Uua: nat,Uub: list_list_nat] :
      ( aa_list_list_nat_o(aa_nat158606821896287091_nat_o(aTP_Lamm_vb(Uu2),Uua),Uub)
    <=> ( ord_le6045566169113846134st_nat(set_list_nat2(Uub),Uu2)
        & ( size_s3023201423986296836st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_228
tff(fact_9327_ATP_Olambda__229,axiom,
    ! [Uu2: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_ug(Uu2),Uua),Uub)
    <=> ( aa_set_o_o(ord_less_eq_set_o(aa_list_o_set_o(set_o2,Uub)),Uu2)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_229
tff(fact_9328_ATP_Olambda__230,axiom,
    ! [Uu2: set_set_nat,Uua: nat,Uub: list_set_nat] :
      ( aa_list_set_nat_o(aa_nat2247512698059773005_nat_o(aTP_Lamm_vc(Uu2),Uua),Uub)
    <=> ( aa_set_set_nat_o(ord_le6893508408891458716et_nat(set_set_nat2(Uub)),Uu2)
        & ( size_s3254054031482475050et_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_230
tff(fact_9329_ATP_Olambda__231,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_uh(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_list_nat_set_nat(set_nat2,Uub)),Uu2)
        & ( aa_list_nat_nat(size_size_list_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_231
tff(fact_9330_ATP_Olambda__232,axiom,
    ! [Uu2: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_ui(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(aa_list_int_set_int(set_int2,Uub)),Uu2)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_232
tff(fact_9331_ATP_Olambda__233,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_wq(Uu2),Uua),Uub)
    <=> ( ( aa_list_nat_nat(size_size_list_nat,Uub) = Uu2 )
        & ( aa_list_nat_nat(groups4561878855575611511st_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_233
tff(fact_9332_ATP_Olambda__234,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_tf(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_234
tff(fact_9333_ATP_Olambda__235,axiom,
    ! [Uu2: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_wz(Uu2),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),pow_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Uu2)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_235
tff(fact_9334_ATP_Olambda__236,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wo(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),vEBT_set_vebt(Uu2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_236
tff(fact_9335_ATP_Olambda__237,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_wp(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),vEBT_set_vebt(Uu2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_237
tff(fact_9336_ATP_Olambda__238,axiom,
    ! [Uu2: list_VEBT_VEBT,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : ( aa_lis2508312872377092907i_assn(aa_arr7909941341466502432i_assn(aTP_Lamm_wi(Uu2),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(Uua,Uub)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uu2,Uub)) ) ).

% ATP.lambda_238
tff(fact_9337_ATP_Olambda__239,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_te(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_239
tff(fact_9338_ATP_Olambda__240,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wh(Uu2),Uua),Uub) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uub)) ) ).

% ATP.lambda_240
tff(fact_9339_ATP_Olambda__241,axiom,
    ! [Uu2: vEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_eb(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_241
tff(fact_9340_ATP_Olambda__242,axiom,
    ! [Uu2: complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_ed(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_242
tff(fact_9341_ATP_Olambda__243,axiom,
    ! [Uu2: real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_ec(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_243
tff(fact_9342_ATP_Olambda__244,axiom,
    ! [Uu2: list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_ee(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_244
tff(fact_9343_ATP_Olambda__245,axiom,
    ! [Uu2: $o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_ea((Uu2)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu2) )
        | aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_245
tff(fact_9344_ATP_Olambda__246,axiom,
    ! [Uu2: set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_ef(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_246
tff(fact_9345_ATP_Olambda__247,axiom,
    ! [Uu2: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_eg(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_247
tff(fact_9346_ATP_Olambda__248,axiom,
    ! [Uu2: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_eh(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_248
tff(fact_9347_ATP_Olambda__249,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_yf(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_249
tff(fact_9348_ATP_Olambda__250,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_vr(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_250
tff(fact_9349_ATP_Olambda__251,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ye(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_251
tff(fact_9350_ATP_Olambda__252,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_vu(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_252
tff(fact_9351_ATP_Olambda__253,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_vt(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu2),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_253
tff(fact_9352_ATP_Olambda__254,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_vs(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu2),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_254
tff(fact_9353_ATP_Olambda__255,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_be(Uu2),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu2),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_255
tff(fact_9354_ATP_Olambda__256,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bf(Uu2),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu2),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_256
tff(fact_9355_ATP_Olambda__257,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aag(Uu2),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu2) ) ) ).

% ATP.lambda_257
tff(fact_9356_ATP_Olambda__258,axiom,
    ! [Uu2: set_Code_integer,Uua: set_Code_integer,Uub: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(aTP_Lamm_pw(Uu2),Uua),Uub)
    <=> ( minus_2355218937544613996nteger(Uub,Uu2) = minus_2355218937544613996nteger(Uua,Uu2) ) ) ).

% ATP.lambda_258
tff(fact_9357_ATP_Olambda__259,axiom,
    ! [Uu2: set_complex,Uua: set_complex,Uub: set_complex] :
      ( aa_set_complex_o(aa_set8643480048548153067plex_o(aTP_Lamm_pv(Uu2),Uua),Uub)
    <=> ( minus_811609699411566653omplex(Uub,Uu2) = minus_811609699411566653omplex(Uua,Uu2) ) ) ).

% ATP.lambda_259
tff(fact_9358_ATP_Olambda__260,axiom,
    ! [Uu2: set_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_px(Uu2),Uua),Uub)
    <=> ( minus_minus_set_nat(Uub,Uu2) = minus_minus_set_nat(Uua,Uu2) ) ) ).

% ATP.lambda_260
tff(fact_9359_ATP_Olambda__261,axiom,
    ! [Uu2: set_int,Uua: set_int,Uub: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_pu(Uu2),Uua),Uub)
    <=> ( minus_minus_set_int(Uub,Uu2) = minus_minus_set_int(Uua,Uu2) ) ) ).

% ATP.lambda_261
tff(fact_9360_ATP_Olambda__262,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_dt(Uu2),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu2)
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_262
tff(fact_9361_ATP_Olambda__263,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_nd(Uu2),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu2)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_263
tff(fact_9362_ATP_Olambda__264,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_dv(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_264
tff(fact_9363_ATP_Olambda__265,axiom,
    ! [Uu2: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_du(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_265
tff(fact_9364_ATP_Olambda__266,axiom,
    ! [Uu2: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_dw(Uu2),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu2)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_266
tff(fact_9365_ATP_Olambda__267,axiom,
    ! [Uu2: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_dx(Uu2),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu2)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_267
tff(fact_9366_ATP_Olambda__268,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_dy(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_268
tff(fact_9367_ATP_Olambda__269,axiom,
    ! [Uu2: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_dz(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_269
tff(fact_9368_ATP_Olambda__270,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_hw(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_270
tff(fact_9369_ATP_Olambda__271,axiom,
    ! [Uu2: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_hx(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_271
tff(fact_9370_ATP_Olambda__272,axiom,
    ! [Uu2: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_hy(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_272
tff(fact_9371_ATP_Olambda__273,axiom,
    ! [Uu2: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ic((Uu2)),Uua),(Uub))
    <=> ( ( (Uu2)
        <=> (Uub) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_273
tff(fact_9372_ATP_Olambda__274,axiom,
    ! [Uu2: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_hz(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_274
tff(fact_9373_ATP_Olambda__275,axiom,
    ! [Uu2: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ia(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_275
tff(fact_9374_ATP_Olambda__276,axiom,
    ! [Uu2: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ib(Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_276
tff(fact_9375_ATP_Olambda__277,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_id(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_277
tff(fact_9376_ATP_Olambda__278,axiom,
    ! [Uu2: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ie(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_278
tff(fact_9377_ATP_Olambda__279,axiom,
    ! [Uu2: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_if(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_279
tff(fact_9378_ATP_Olambda__280,axiom,
    ! [Uu2: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ij((Uu2)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu2) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_280
tff(fact_9379_ATP_Olambda__281,axiom,
    ! [Uu2: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ig(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_281
tff(fact_9380_ATP_Olambda__282,axiom,
    ! [Uu2: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ih(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_282
tff(fact_9381_ATP_Olambda__283,axiom,
    ! [Uu2: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ii(Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_283
tff(fact_9382_ATP_Olambda__284,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun6272599145770227614VEBT_o(aTP_Lamm_ky(Uu2),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu2)
        & ( aa_VEBT_VEBT_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_284
tff(fact_9383_ATP_Olambda__285,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aTP_Lamm_lk(Uu2),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu2)
        & ( aa_VEBT_VEBT_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_285
tff(fact_9384_ATP_Olambda__286,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Co1554742298434871494uint32,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8926496376162642256eger_o(aTP_Lamm_li(Uu2),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu2)
        & ( aa_Cod8401701657498852871uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_286
tff(fact_9385_ATP_Olambda__287,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_uint32,Uub: complex] :
      ( aa_complex_o(aa_fun436548534304416686plex_o(aTP_Lamm_lg(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        & ( aa_complex_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_287
tff(fact_9386_ATP_Olambda__288,axiom,
    ! [Uu2: set_real,Uua: fun_real_uint32,Uub: real] :
      ( aa_real_o(aa_fun7556296611544133298real_o(aTP_Lamm_la(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & ( aa_real_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_288
tff(fact_9387_ATP_Olambda__289,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_lm(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_289
tff(fact_9388_ATP_Olambda__290,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_uint32,Uub: nat] :
      ( aa_nat_o(aa_fun1182053767699555178_nat_o(aTP_Lamm_lc(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_290
tff(fact_9389_ATP_Olambda__291,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_lo(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_291
tff(fact_9390_ATP_Olambda__292,axiom,
    ! [Uu2: set_int,Uua: fun_int_uint32,Uub: int] :
      ( aa_int_o(aa_fun6992502331602369202_int_o(aTP_Lamm_le(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & ( aa_int_uint32(Uua,Uub) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_292
tff(fact_9391_ATP_Olambda__293,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_lq(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_293
tff(fact_9392_ATP_Olambda__294,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5142337294461892954VEBT_o(aTP_Lamm_me(Uu2),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu2)
        & ( aa_VEBT_VEBT_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_294
tff(fact_9393_ATP_Olambda__295,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5889122065779655690VEBT_o(aTP_Lamm_ls(Uu2),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu2)
        & ( aa_VEBT_VEBT_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_295
tff(fact_9394_ATP_Olambda__296,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Code_integer_rat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun6395433285121891940eger_o(aTP_Lamm_mc(Uu2),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu2)
        & ( aa_Code_integer_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_296
tff(fact_9395_ATP_Olambda__297,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_rat,Uub: complex] :
      ( aa_complex_o(aa_fun5099116602557847362plex_o(aTP_Lamm_ma(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        & ( aa_complex_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_297
tff(fact_9396_ATP_Olambda__298,axiom,
    ! [Uu2: set_real,Uua: fun_real_assn,Uub: real] :
      ( aa_real_o(aa_fun7580339382299851254real_o(aTP_Lamm_mg(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & ( aa_real_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_298
tff(fact_9397_ATP_Olambda__299,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_lu(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_299
tff(fact_9398_ATP_Olambda__300,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_assn,Uub: nat] :
      ( aa_nat_o(aa_fun7183233715851567790_nat_o(aTP_Lamm_mi(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_300
tff(fact_9399_ATP_Olambda__301,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_lw(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_301
tff(fact_9400_ATP_Olambda__302,axiom,
    ! [Uu2: set_int,Uua: fun_int_assn,Uub: int] :
      ( aa_int_o(aa_fun202081646129719286_int_o(aTP_Lamm_mk(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & ( aa_int_assn(Uua,Uub) != one_one_assn ) ) ) ).

% ATP.lambda_302
tff(fact_9401_ATP_Olambda__303,axiom,
    ! [Uu2: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_ly(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & ( aa_int_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_303
tff(fact_9402_ATP_Olambda__304,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : ( aa_o_assn(aa_VEB6687300313434771872o_assn(aTP_Lamm_vl(Uu2),Uua),(Uub)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu2),Uua)),aa_o_assn(pure_assn,(Uub) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Uu2))) ) ).

% ATP.lambda_304
tff(fact_9403_ATP_Olambda__305,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : ( aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_vo(Uu2),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu2),Uua)),aa_o_assn(pure_assn,Uub = vEBT_vebt_mint(Uu2))) ) ).

% ATP.lambda_305
tff(fact_9404_ATP_Olambda__306,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : ( aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_vp(Uu2),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu2),Uua)),aa_o_assn(pure_assn,Uub = vEBT_vebt_maxt(Uu2))) ) ).

% ATP.lambda_306
tff(fact_9405_ATP_Olambda__307,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_fm(Uu2),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu2)
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_307
tff(fact_9406_ATP_Olambda__308,axiom,
    ! [Uu2: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_fo(Uu2),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu2)
        & ~ aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_308
tff(fact_9407_ATP_Olambda__309,axiom,
    ! [Uu2: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_fn(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu2)
        & ~ aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_309
tff(fact_9408_ATP_Olambda__310,axiom,
    ! [Uu2: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_fp(Uu2),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu2)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_310
tff(fact_9409_ATP_Olambda__311,axiom,
    ! [Uu2: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_fq(Uu2),Uua),Uub)
    <=> ( aa_set_set_nat_o(member_set_nat(Uub),Uu2)
        & ~ aa_set_set_nat_o(member_set_nat(Uub),Uua) ) ) ).

% ATP.lambda_311
tff(fact_9410_ATP_Olambda__312,axiom,
    ! [Uu2: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_fs(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu2)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_312
tff(fact_9411_ATP_Olambda__313,axiom,
    ! [Uu2: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_fr(Uu2),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu2)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_313
tff(fact_9412_ATP_Olambda__314,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_we(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu2) ) ).

% ATP.lambda_314
tff(fact_9413_ATP_Olambda__315,axiom,
    ! [Uu2: set_Pr1773385645901665561uint32,Uua: uint32,Uub: uint32] :
      ( aa_uint32_o(aa_uin5924812838015166427nt32_o(aTP_Lamm_kh(Uu2),Uua),Uub)
    <=> member8027108493173000802uint32(produc1400373151660368625uint32(Uua,Uub),Uu2) ) ).

% ATP.lambda_315
tff(fact_9414_ATP_Olambda__316,axiom,
    ! [Uu2: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ki(Uu2),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub),Uu2) ) ).

% ATP.lambda_316
tff(fact_9415_ATP_Olambda__317,axiom,
    ! [Uu2: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_kj(Uu2),Uua),Uub)
    <=> member5262025264175285858nt_int(product_Pair_int_int(Uua,Uub),Uu2) ) ).

% ATP.lambda_317
tff(fact_9416_ATP_Olambda__318,axiom,
    ! [Uu2: set_Pr1312406793078187350et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_kk(Uu2),Uua),Uub)
    <=> member4513763875858039863et_nat(produc6017135503889078682et_nat(Uua,Uub),Uu2) ) ).

% ATP.lambda_318
tff(fact_9417_ATP_Olambda__319,axiom,
    ! [Uu2: set_Pr4040394263813967220et_nat,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc8799533983703829792et_nat] :
      ( aa_Pro5643009357358181178_nat_o(aa_fun912680268133416158_nat_o(aTP_Lamm_kl(Uu2),Uua),Uub)
    <=> member9174161442226468181et_nat(produc7291855603632349880et_nat(Uua,Uub),Uu2) ) ).

% ATP.lambda_319
tff(fact_9418_ATP_Olambda__320,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_vv(Uu2),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu2) = Uua ) ) ).

% ATP.lambda_320
tff(fact_9419_ATP_Olambda__321,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_vx(Uu2),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_321
tff(fact_9420_ATP_Olambda__322,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_on(Uu2),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),ring_1_Ints_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu2),Uub)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uub),Uua) ) ) ).

% ATP.lambda_322
tff(fact_9421_ATP_Olambda__323,axiom,
    ! [Uu2: rat,Uua: rat,Uub: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_oo(Uu2),Uua),Uub)
    <=> ( aa_set_rat_o(member_rat(Uub),ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uu2),Uub)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uub),Uua) ) ) ).

% ATP.lambda_323
tff(fact_9422_ATP_Olambda__324,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yx(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ) ).

% ATP.lambda_324
tff(fact_9423_ATP_Olambda__325,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xy(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ) ).

% ATP.lambda_325
tff(fact_9424_ATP_Olambda__326,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wb(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ) ).

% ATP.lambda_326
tff(fact_9425_ATP_Olambda__327,axiom,
    ! [Uu2: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_km(Uu2),Uua),Uub)
    <=> ( aa_nat_o(Uu2,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_327
tff(fact_9426_ATP_Olambda__328,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vq(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_328
tff(fact_9427_ATP_Olambda__329,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : ( aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ze(Uu2),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ) ).

% ATP.lambda_329
tff(fact_9428_ATP_Olambda__330,axiom,
    ! [Uu2: fun_VEBT_VEBT_nat,Uua: fun_VEBT_VEBT_nat,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aTP_Lamm_op(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_VEBT_VEBT_nat(Uua,Uub)),aa_VEBT_VEBT_nat(Uu2,Uub)) ) ).

% ATP.lambda_330
tff(fact_9429_ATP_Olambda__331,axiom,
    ! [Uu2: fun_real_nat,Uua: fun_real_nat,Uub: real] : ( aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_kf(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu2,Uub)) ) ).

% ATP.lambda_331
tff(fact_9430_ATP_Olambda__332,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : ( aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_oq(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uua,Uub)),aa_set_nat_nat(Uu2,Uub)) ) ).

% ATP.lambda_332
tff(fact_9431_ATP_Olambda__333,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_kg(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu2,Uub)) ) ).

% ATP.lambda_333
tff(fact_9432_ATP_Olambda__334,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_int_nat,Uub: int] : ( aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ke(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu2,Uub)) ) ).

% ATP.lambda_334
tff(fact_9433_ATP_Olambda__335,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : ( aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ym(Uu2),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ) ).

% ATP.lambda_335
tff(fact_9434_ATP_Olambda__336,axiom,
    ! [Uu2: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_ax(Uu2),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu2,Uub)
        | aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_336
tff(fact_9435_ATP_Olambda__337,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_aw(Uu2),Uua),Uub)
    <=> ( aa_complex_o(Uu2,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_337
tff(fact_9436_ATP_Olambda__338,axiom,
    ! [Uu2: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_as(Uu2),Uua),Uub)
    <=> ( aa_list_nat_o(Uu2,Uub)
        | aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_338
tff(fact_9437_ATP_Olambda__339,axiom,
    ! [Uu2: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_at(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(Uu2,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_339
tff(fact_9438_ATP_Olambda__340,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_au(Uu2),Uua),Uub)
    <=> ( aa_nat_o(Uu2,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_340
tff(fact_9439_ATP_Olambda__341,axiom,
    ! [Uu2: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_av(Uu2),Uua),Uub)
    <=> ( aa_int_o(Uu2,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_341
tff(fact_9440_ATP_Olambda__342,axiom,
    ! [Uu2: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_bd(Uu2),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu2,Uub)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_342
tff(fact_9441_ATP_Olambda__343,axiom,
    ! [Uu2: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_bc(Uu2),Uua),Uub)
    <=> ( aa_complex_o(Uu2,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_343
tff(fact_9442_ATP_Olambda__344,axiom,
    ! [Uu2: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ay(Uu2),Uua),Uub)
    <=> ( aa_list_nat_o(Uu2,Uub)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_344
tff(fact_9443_ATP_Olambda__345,axiom,
    ! [Uu2: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_az(Uu2),Uua),Uub)
    <=> ( aa_set_nat_o(Uu2,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_345
tff(fact_9444_ATP_Olambda__346,axiom,
    ! [Uu2: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ba(Uu2),Uua),Uub)
    <=> ( aa_nat_o(Uu2,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_346
tff(fact_9445_ATP_Olambda__347,axiom,
    ! [Uu2: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bb(Uu2),Uua),Uub)
    <=> ( aa_int_o(Uu2,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_347
tff(fact_9446_ATP_Olambda__348,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_kn(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu2,Uub)),Uua) ) ).

% ATP.lambda_348
tff(fact_9447_ATP_Olambda__349,axiom,
    ! [Uu2: nat,Uua: fun_real_real,Uub: real] : ( aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zr(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu2) ) ).

% ATP.lambda_349
tff(fact_9448_ATP_Olambda__350,axiom,
    ! [Uu2: fun_Co3960896333048504454r_real,Uua: real,Uub: code_integer] : ( aa_Code_integer_real(aa_rea8190550742337927152r_real(aTP_Lamm_tr(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(Uu2,Uub)),Uua) ) ).

% ATP.lambda_350
tff(fact_9449_ATP_Olambda__351,axiom,
    ! [Uu2: fun_Code_integer_rat,Uua: rat,Uub: code_integer] : ( aa_Code_integer_rat(aa_rat6553461586499541832er_rat(aTP_Lamm_tt(Uu2),Uua),Uub) = aa_rat_rat(plus_plus_rat(aa_Code_integer_rat(Uu2,Uub)),Uua) ) ).

% ATP.lambda_351
tff(fact_9450_ATP_Olambda__352,axiom,
    ! [Uu2: fun_Code_integer_int,Uua: int,Uub: code_integer] : ( aa_Code_integer_int(aa_int1390634949194299888er_int(aTP_Lamm_tv(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(Uu2,Uub)),Uua) ) ).

% ATP.lambda_352
tff(fact_9451_ATP_Olambda__353,axiom,
    ! [Uu2: fun_complex_real,Uua: real,Uub: complex] : ( aa_complex_real(aa_rea4562580341872191937x_real(aTP_Lamm_tq(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uu2,Uub)),Uua) ) ).

% ATP.lambda_353
tff(fact_9452_ATP_Olambda__354,axiom,
    ! [Uu2: fun_complex_rat,Uua: rat,Uub: complex] : ( aa_complex_rat(aa_rat15117248669498777ex_rat(aTP_Lamm_ts(Uu2),Uua),Uub) = aa_rat_rat(plus_plus_rat(aa_complex_rat(Uu2,Uub)),Uua) ) ).

% ATP.lambda_354
tff(fact_9453_ATP_Olambda__355,axiom,
    ! [Uu2: fun_complex_int,Uua: int,Uub: complex] : ( aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_tu(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(Uu2,Uub)),Uua) ) ).

% ATP.lambda_355
tff(fact_9454_ATP_Olambda__356,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tw(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu2,Uub)),Uua) ) ).

% ATP.lambda_356
tff(fact_9455_ATP_Olambda__357,axiom,
    ! [Uu2: fun_nat_rat,Uua: rat,Uub: nat] : ( aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_tx(Uu2),Uua),Uub) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu2,Uub)),Uua) ) ).

% ATP.lambda_357
tff(fact_9456_ATP_Olambda__358,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : ( aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ty(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu2,Uub)),Uua) ) ).

% ATP.lambda_358
tff(fact_9457_ATP_Olambda__359,axiom,
    ! [Uu2: fun_int_real,Uua: real,Uub: int] : ( aa_int_real(aa_real_fun_int_real(aTP_Lamm_tz(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uu2,Uub)),Uua) ) ).

% ATP.lambda_359
tff(fact_9458_ATP_Olambda__360,axiom,
    ! [Uu2: fun_real_real,Uua: real,Uub: real] : ( aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_yl(Uu2),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu2,Uub)),Uua) ) ).

% ATP.lambda_360
tff(fact_9459_ATP_Olambda__361,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ei(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_361
tff(fact_9460_ATP_Olambda__362,axiom,
    ! [Uu2: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ek(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_362
tff(fact_9461_ATP_Olambda__363,axiom,
    ! [Uu2: list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_el(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_363
tff(fact_9462_ATP_Olambda__364,axiom,
    ! [Uu2: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ej((Uu2)),Uua),(Uub))
    <=> ( ~ ( (Uub)
          <=> (Uu2) )
       => aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_364
tff(fact_9463_ATP_Olambda__365,axiom,
    ! [Uu2: set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_em(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_365
tff(fact_9464_ATP_Olambda__366,axiom,
    ! [Uu2: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_en(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_366
tff(fact_9465_ATP_Olambda__367,axiom,
    ! [Uu2: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_eo(Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_367
tff(fact_9466_ATP_Olambda__368,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : ( aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_tm(Uu2),Uua),Uub) = aa_nat_complex(power_power_complex(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ) ).

% ATP.lambda_368
tff(fact_9467_ATP_Olambda__369,axiom,
    ! [Uu2: uint32,Uua: nat,Uub: nat] : ( aa_nat_uint32(aa_nat6935575117487981255uint32(aTP_Lamm_tn(Uu2),Uua),Uub) = aa_nat_uint32(power_power_uint32(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ) ).

% ATP.lambda_369
tff(fact_9468_ATP_Olambda__370,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : ( aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_tp(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ) ).

% ATP.lambda_370
tff(fact_9469_ATP_Olambda__371,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : ( aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_tl(Uu2),Uua),Uub) = aa_nat_rat(power_power_rat(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ) ).

% ATP.lambda_371
tff(fact_9470_ATP_Olambda__372,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : ( aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_to(Uu2),Uua),Uub) = aa_nat_int(power_power_int(Uu2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ) ).

% ATP.lambda_372
tff(fact_9471_ATP_Olambda__373,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : ( aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zj(Uu2),Uua),Uub) = aa_real_real(divide_divide_real(Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ) ).

% ATP.lambda_373
tff(fact_9472_ATP_Olambda__374,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xa(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ) ).

% ATP.lambda_374
tff(fact_9473_ATP_Olambda__375,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_zs(Uu2),Uua),Uub)
    <=> aa_set_real_o(member_real(Uub),set_or1633881224788618240n_real(Uu2,Uua)) ) ).

% ATP.lambda_375
tff(fact_9474_ATP_Olambda__376,axiom,
    ! [Uu2: fun_Code_integer_nat,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_qt(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Code_integer_nat(Uu2,Uub)),aa_Code_integer_nat(Uu2,Uub))) ) ).

% ATP.lambda_376
tff(fact_9475_ATP_Olambda__377,axiom,
    ! [Uu2: fun_complex_nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_qs(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_complex_nat(Uu2,Uub)),aa_complex_nat(Uu2,Uub))) ) ).

% ATP.lambda_377
tff(fact_9476_ATP_Olambda__378,axiom,
    ! [Uu2: fun_list_nat_nat,Uua: list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(aTP_Lamm_qo(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_list_nat_nat(Uu2,Uub)),aa_list_nat_nat(Uu2,Uub))) ) ).

% ATP.lambda_378
tff(fact_9477_ATP_Olambda__379,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_qp(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_set_nat_nat(Uu2,Uub)),aa_set_nat_nat(Uu2,Uub))) ) ).

% ATP.lambda_379
tff(fact_9478_ATP_Olambda__380,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_qq(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uu2,Uub))) ) ).

% ATP.lambda_380
tff(fact_9479_ATP_Olambda__381,axiom,
    ! [Uu2: fun_int_nat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qr(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_int_nat(Uu2,Uub)),aa_int_nat(Uu2,Uub))) ) ).

% ATP.lambda_381
tff(fact_9480_ATP_Olambda__382,axiom,
    ! [Uu2: fun_Code_integer_nat,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_mx(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_nat(Uu2,Uub),zero_zero_nat)) ) ).

% ATP.lambda_382
tff(fact_9481_ATP_Olambda__383,axiom,
    ! [Uu2: fun_complex_nat,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_mv(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_complex_o(Uua,Uub),aa_complex_nat(Uu2,Uub),zero_zero_nat)) ) ).

% ATP.lambda_383
tff(fact_9482_ATP_Olambda__384,axiom,
    ! [Uu2: fun_list_nat_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_mn(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_list_nat_o(Uua,Uub),aa_list_nat_nat(Uu2,Uub),zero_zero_nat)) ) ).

% ATP.lambda_384
tff(fact_9483_ATP_Olambda__385,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_mp(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_set_nat_o(Uua,Uub),aa_set_nat_nat(Uu2,Uub),zero_zero_nat)) ) ).

% ATP.lambda_385
tff(fact_9484_ATP_Olambda__386,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_mr(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu2,Uub),zero_zero_nat)) ) ).

% ATP.lambda_386
tff(fact_9485_ATP_Olambda__387,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_mt(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu2,Uub),zero_zero_nat)) ) ).

% ATP.lambda_387
tff(fact_9486_ATP_Olambda__388,axiom,
    ! [Uu2: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun4054834855839773292eger_o(aTP_Lamm_rz(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_integer_nat(Uu2,Uub)),aa_Code_integer_nat(Uua,Uub))) ) ).

% ATP.lambda_388
tff(fact_9487_ATP_Olambda__389,axiom,
    ! [Uu2: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] :
      ( aa_complex_o(aa_fun8723867720292580170plex_o(aTP_Lamm_ry(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uu2,Uub)),aa_complex_nat(Uua,Uub))) ) ).

% ATP.lambda_389
tff(fact_9488_ATP_Olambda__390,axiom,
    ! [Uu2: fun_list_nat_nat,Uua: fun_list_nat_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun7915899554530022_nat_o(aTP_Lamm_ru(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_list_nat_nat(Uu2,Uub)),aa_list_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_390
tff(fact_9489_ATP_Olambda__391,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun1332161875654541722_nat_o(aTP_Lamm_rv(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(Uu2,Uub)),aa_set_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_391
tff(fact_9490_ATP_Olambda__392,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_rw(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_392
tff(fact_9491_ATP_Olambda__393,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_rx(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uu2,Uub)),aa_int_nat(Uua,Uub))) ) ).

% ATP.lambda_393
tff(fact_9492_ATP_Olambda__394,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : ( aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_hl(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ) ).

% ATP.lambda_394
tff(fact_9493_ATP_Olambda__395,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hk(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ) ).

% ATP.lambda_395
tff(fact_9494_ATP_Olambda__396,axiom,
    ! [Uu2: fun_set_nat_nat,Uua: fun_nat_set_nat,Uub: nat] : ( aa_nat_nat(aa_fun4820382421437094446at_nat(aTP_Lamm_gl(Uu2),Uua),Uub) = aa_set_nat_nat(Uu2,aa_nat_set_nat(Uua,Uub)) ) ).

% ATP.lambda_396
tff(fact_9495_ATP_Olambda__397,axiom,
    ! [Uu2: fun_set_nat_int,Uua: fun_nat_set_nat,Uub: nat] : ( aa_nat_int(aa_fun642531401927897738at_int(aTP_Lamm_gm(Uu2),Uua),Uub) = aa_set_nat_int(Uu2,aa_nat_set_nat(Uua,Uub)) ) ).

% ATP.lambda_397
tff(fact_9496_ATP_Olambda__398,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_gd(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_398
tff(fact_9497_ATP_Olambda__399,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_int_nat,Uub: int] : ( aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ge(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_int_nat(Uua,Uub)) ) ).

% ATP.lambda_399
tff(fact_9498_ATP_Olambda__400,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_nat,Uub: nat] : ( aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_gf(Uu2),Uua),Uub) = aa_nat_int(Uu2,aa_nat_nat(Uua,Uub)) ) ).

% ATP.lambda_400
tff(fact_9499_ATP_Olambda__401,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_int_nat,Uub: int] : ( aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_gg(Uu2),Uua),Uub) = aa_nat_int(Uu2,aa_int_nat(Uua,Uub)) ) ).

% ATP.lambda_401
tff(fact_9500_ATP_Olambda__402,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_nat_int,Uub: nat] : ( aa_nat_nat(aa_fun519522104774525396at_nat(aTP_Lamm_gh(Uu2),Uua),Uub) = aa_int_nat(Uu2,aa_nat_int(Uua,Uub)) ) ).

% ATP.lambda_402
tff(fact_9501_ATP_Olambda__403,axiom,
    ! [Uu2: fun_int_nat,Uua: fun_int_int,Uub: int] : ( aa_int_nat(aa_fun1234824742305470092nt_nat(aTP_Lamm_gi(Uu2),Uua),Uub) = aa_int_nat(Uu2,aa_int_int(Uua,Uub)) ) ).

% ATP.lambda_403
tff(fact_9502_ATP_Olambda__404,axiom,
    ! [Uu2: fun_int_int,Uua: fun_nat_int,Uub: nat] : ( aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_gj(Uu2),Uua),Uub) = aa_int_int(Uu2,aa_nat_int(Uua,Uub)) ) ).

% ATP.lambda_404
tff(fact_9503_ATP_Olambda__405,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : ( aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_gk(Uu2),Uua),Uub) = aa_int_int(Uu2,aa_int_int(Uua,Uub)) ) ).

% ATP.lambda_405
tff(fact_9504_ATP_Olambda__406,axiom,
    ! [Uu2: fun_VE9219535225822561060nteger,Uua: fun_Co1554742298434871494uint32,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_uint32(aa_fun1617233732993516207uint32(aTP_Lamm_ri(Uu2),Uua),Uub) = aa_Cod8401701657498852871uint32(Uua,aa_VEB5935388676523804373nteger(Uu2,Uub)) ) ).

% ATP.lambda_406
tff(fact_9505_ATP_Olambda__407,axiom,
    ! [Uu2: fun_VE5569070535936029429omplex,Uua: fun_complex_uint32,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_uint32(aa_fun4101646971871848350uint32(aTP_Lamm_rb(Uu2),Uua),Uub) = aa_complex_uint32(Uua,aa_VEBT_VEBT_complex(Uu2,Uub)) ) ).

% ATP.lambda_407
tff(fact_9506_ATP_Olambda__408,axiom,
    ! [Uu2: fun_VEBT_VEBT_nat,Uua: fun_nat_uint32,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_uint32(aa_fun2276759601835508476uint32(aTP_Lamm_qu(Uu2),Uua),Uub) = aa_nat_uint32(Uua,aa_VEBT_VEBT_nat(Uu2,Uub)) ) ).

% ATP.lambda_408
tff(fact_9507_ATP_Olambda__409,axiom,
    ! [Uu2: fun_li486744618827423812nteger,Uua: fun_Co1554742298434871494uint32,Uub: literal] : ( aa_literal_uint32(aa_fun7307945177061598671uint32(aTP_Lamm_ro(Uu2),Uua),Uub) = aa_Cod8401701657498852871uint32(Uua,aa_lit3353659836003880949nteger(Uu2,Uub)) ) ).

% ATP.lambda_409
tff(fact_9508_ATP_Olambda__410,axiom,
    ! [Uu2: fun_literal_complex,Uua: fun_complex_uint32,Uub: literal] : ( aa_literal_uint32(aa_fun794436630273323070uint32(aTP_Lamm_rf(Uu2),Uua),Uub) = aa_complex_uint32(Uua,aa_literal_complex(Uu2,Uub)) ) ).

% ATP.lambda_410
tff(fact_9509_ATP_Olambda__411,axiom,
    ! [Uu2: fun_literal_nat,Uua: fun_nat_uint32,Uub: literal] : ( aa_literal_uint32(aa_fun6956465913453778588uint32(aTP_Lamm_qy(Uu2),Uua),Uub) = aa_nat_uint32(Uua,aa_literal_nat(Uu2,Uub)) ) ).

% ATP.lambda_411
tff(fact_9510_ATP_Olambda__412,axiom,
    ! [Uu2: fun_re6978948838062957830nteger,Uua: fun_Co1554742298434871494uint32,Uub: real] : ( aa_real_uint32(aa_fun94539430827241881uint32(aTP_Lamm_rl(Uu2),Uua),Uub) = aa_Cod8401701657498852871uint32(Uua,aa_real_Code_integer(Uu2,Uub)) ) ).

% ATP.lambda_412
tff(fact_9511_ATP_Olambda__413,axiom,
    ! [Uu2: fun_real_complex,Uua: fun_complex_uint32,Uub: real] : ( aa_real_uint32(aa_fun6757232495900413034uint32(aTP_Lamm_rd(Uu2),Uua),Uub) = aa_complex_uint32(Uua,aa_real_complex(Uu2,Uub)) ) ).

% ATP.lambda_413
tff(fact_9512_ATP_Olambda__414,axiom,
    ! [Uu2: fun_real_nat,Uua: fun_nat_uint32,Uub: real] : ( aa_real_uint32(aa_fun5130233130708504716uint32(aTP_Lamm_qw(Uu2),Uua),Uub) = aa_nat_uint32(Uua,aa_real_nat(Uu2,Uub)) ) ).

% ATP.lambda_414
tff(fact_9513_ATP_Olambda__415,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_uint32,Uub: nat] : ( aa_nat_uint32(aa_fun7322500112117231024uint32(aTP_Lamm_rr(Uu2),Uua),Uub) = aa_nat_uint32(Uua,aa_nat_nat(Uu2,Uub)) ) ).

% ATP.lambda_415
tff(fact_9514_ATP_Olambda__416,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : ( aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_xj(Uu2),Uua),Uub) = aa_o_assn(pure_assn,Uub = aa_nat_nat(vEBT_VEBT_high(Uu2),Uua)) ) ).

% ATP.lambda_416
tff(fact_9515_ATP_Olambda__417,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : ( aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_xi(Uu2),Uua),Uub) = aa_o_assn(pure_assn,Uub = aa_nat_nat(vEBT_VEBT_low(Uu2),Uua)) ) ).

% ATP.lambda_417
tff(fact_9516_ATP_Olambda__418,axiom,
    ! [Uu2: set_int,Uua: fun_VE4235606543851232980_int_o,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aa_fun5588456476524577925BT_nat(aTP_Lamm_pt(Uu2),Uua),Uub) = finite_card_int(collect_int(aa_VEB493578126231709133_int_o(aa_fun8464664609713939430_int_o(aTP_Lamm_ps(Uu2),Uua),Uub))) ) ).

% ATP.lambda_418
tff(fact_9517_ATP_Olambda__419,axiom,
    ! [Uu2: set_literal,Uua: fun_li4177005666941401710eral_o,Uub: literal] : ( aa_literal_nat(aa_fun8110944661589842437al_nat(aTP_Lamm_pr(Uu2),Uua),Uub) = finite_card_literal(collect_literal(aa_lit5986169069484365087eral_o(aa_fun6827806182750346556eral_o(aTP_Lamm_pq(Uu2),Uua),Uub))) ) ).

% ATP.lambda_419
tff(fact_9518_ATP_Olambda__420,axiom,
    ! [Uu2: set_real,Uua: fun_li5129792558484764084real_o,Uub: literal] : ( aa_literal_nat(aa_fun202459941351508933al_nat(aTP_Lamm_po(Uu2),Uua),Uub) = finite_card_real(collect_real(aa_lit1368651074454191789real_o(aa_fun6987840782327099302real_o(aTP_Lamm_pn(Uu2),Uua),Uub))) ) ).

% ATP.lambda_420
tff(fact_9519_ATP_Olambda__421,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_li8107907266037299598VEBT_o,Uub: literal] : ( aa_literal_nat(aa_fun1876825894470965861al_nat(aTP_Lamm_pl(Uu2),Uua),Uub) = finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_lit1284886718724587071VEBT_o(aa_fun839249340096772540VEBT_o(aTP_Lamm_pk(Uu2),Uua),Uub))) ) ).

% ATP.lambda_421
tff(fact_9520_ATP_Olambda__422,axiom,
    ! [Uu2: set_literal,Uua: fun_re3850508403959675952eral_o,Uub: real] : ( aa_real_nat(aa_fun705237418523604071al_nat(aTP_Lamm_pi(Uu2),Uua),Uub) = finite_card_literal(collect_literal(aa_rea1838623199774720753eral_o(aa_fun1820749043261144222eral_o(aTP_Lamm_ph(Uu2),Uua),Uub))) ) ).

% ATP.lambda_422
tff(fact_9521_ATP_Olambda__423,axiom,
    ! [Uu2: set_literal,Uua: fun_VE8371440389858774350eral_o,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aa_fun3014642116068820613BT_nat(aTP_Lamm_pf(Uu2),Uua),Uub) = finite_card_literal(collect_literal(aa_VEB1719478621182999551eral_o(aa_fun5643091140434266556eral_o(aTP_Lamm_pe(Uu2),Uua),Uub))) ) ).

% ATP.lambda_423
tff(fact_9522_ATP_Olambda__424,axiom,
    ! [Uu2: set_real,Uua: fun_real_fun_real_o,Uub: real] : ( aa_real_nat(aa_fun3429493660699118127al_nat(aTP_Lamm_pc(Uu2),Uua),Uub) = finite_card_real(collect_real(aa_real_fun_real_o(aa_fun4467907209059791804real_o(aTP_Lamm_pb(Uu2),Uua),Uub))) ) ).

% ATP.lambda_424
tff(fact_9523_ATP_Olambda__425,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_re7829509182167823952VEBT_o,Uub: real] : ( aa_real_nat(aa_fun1148221025137894919al_nat(aTP_Lamm_oz(Uu2),Uua),Uub) = finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_rea6225614514526338833VEBT_o(aa_fun2719675890530006366VEBT_o(aTP_Lamm_oy(Uu2),Uua),Uub))) ) ).

% ATP.lambda_425
tff(fact_9524_ATP_Olambda__426,axiom,
    ! [Uu2: set_real,Uua: fun_VE3823611782367660756real_o,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aa_fun2935073076783396613BT_nat(aTP_Lamm_ow(Uu2),Uua),Uub) = finite_card_real(collect_real(aa_VEB641519613517496269real_o(aa_fun5792667679087400038real_o(aTP_Lamm_ov(Uu2),Uua),Uub))) ) ).

% ATP.lambda_426
tff(fact_9525_ATP_Olambda__427,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE2962686792283380846VEBT_o,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aa_fun2100738023942440677BT_nat(aTP_Lamm_ot(Uu2),Uua),Uub) = finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_VEB1521802747257091359VEBT_o(aa_fun8771395773580142140VEBT_o(aTP_Lamm_os(Uu2),Uua),Uub))) ) ).

% ATP.lambda_427
tff(fact_9526_ATP_Olambda__428,axiom,
    ! [Uu2: set_nat,Uua: fun_VE6480751511521678896_nat_o,Uub: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aa_fun3621025987637128233BT_nat(aTP_Lamm_td(Uu2),Uua),Uub) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_VEB2314620471664397097_nat_o(aa_fun1998352179508271774_nat_o(aTP_Lamm_ik(Uu2),Uua),Uub))) ) ).

% ATP.lambda_428
tff(fact_9527_ATP_Olambda__429,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_uq(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_VEBT_VEBT_real(Uua,Uuc),aa_VEBT_VEBT_real(Uub,Uuc)) ) ).

% ATP.lambda_429
tff(fact_9528_ATP_Olambda__430,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_uu(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_VEBT_VEBT_rat(Uua,Uuc),aa_VEBT_VEBT_rat(Uub,Uuc)) ) ).

% ATP.lambda_430
tff(fact_9529_ATP_Olambda__431,axiom,
    ! [Uu2: vEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: fun_VEBT_VEBT_nat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_uy(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_VEBT_VEBT_nat(Uua,Uuc),aa_VEBT_VEBT_nat(Uub,Uuc)) ) ).

% ATP.lambda_431
tff(fact_9530_ATP_Olambda__432,axiom,
    ! [Uu2: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: fun_Co3960896333048504454r_real,Uuc: code_integer] :
      ( aa_Code_integer_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_ut(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_Code_integer_real(Uua,Uuc),aa_Code_integer_real(Uub,Uuc)) ) ).

% ATP.lambda_432
tff(fact_9531_ATP_Olambda__433,axiom,
    ! [Uu2: code_integer,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      ( aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_ux(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_Code_integer_rat(Uua,Uuc),aa_Code_integer_rat(Uub,Uuc)) ) ).

% ATP.lambda_433
tff(fact_9532_ATP_Olambda__434,axiom,
    ! [Uu2: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_us(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ) ).

% ATP.lambda_434
tff(fact_9533_ATP_Olambda__435,axiom,
    ! [Uu2: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      ( aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_uw(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ) ).

% ATP.lambda_435
tff(fact_9534_ATP_Olambda__436,axiom,
    ! [Uu2: real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_real(aa_fun6946558556880649704l_real(aa_fun5414376518349398074l_real(aTP_Lamm_ur(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_real_real(Uua,Uuc),aa_real_real(Uub,Uuc)) ) ).

% ATP.lambda_436
tff(fact_9535_ATP_Olambda__437,axiom,
    ! [Uu2: real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_rat(aa_fun2468082234287226944al_rat(aa_fun1537150169341160510al_rat(aTP_Lamm_uv(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_real_rat(Uua,Uuc),aa_real_rat(Uub,Uuc)) ) ).

% ATP.lambda_437
tff(fact_9536_ATP_Olambda__438,axiom,
    ! [Uu2: real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      ( aa_real_nat(aa_fun211602392931233840al_nat(aa_fun1500080954902701606al_nat(aTP_Lamm_uz(Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa_real_nat(Uua,Uuc),aa_real_nat(Uub,Uuc)) ) ).

% ATP.lambda_438
tff(fact_9537_ATP_Olambda__439,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co8975588572093281791plex_o,Uuc: complex] : ( aa_complex_complex(aa_fun7787660480616508700omplex(aa_fun963654016877434554omplex(aTP_Lamm_kd(Uu2),Uua),Uub),Uuc) = groups8024822376189712711omplex(aa_com7552888802784664052omplex(aTP_Lamm_kb(Uua),Uuc),collect_Code_integer(aa_com4877492900066995176eger_o(aa_fun3025727905316630780eger_o(aTP_Lamm_kc(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_439
tff(fact_9538_ATP_Olambda__440,axiom,
    ! [Uu2: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: complex] : ( aa_complex_complex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_jy(Uu2),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_jw(Uua),Uuc),collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_jx(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_440
tff(fact_9539_ATP_Olambda__441,axiom,
    ! [Uu2: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : ( aa_complex_complex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_jt(Uu2),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_jr(Uua),Uuc),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_js(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_441
tff(fact_9540_ATP_Olambda__442,axiom,
    ! [Uu2: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: complex] : ( aa_complex_complex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_jo(Uu2),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_jn(Uua),Uuc),collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_fy(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_442
tff(fact_9541_ATP_Olambda__443,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE7631951351884037842plex_o,Uuc: complex] : ( aa_complex_complex(aa_fun2592672125884331625omplex(aa_fun8556514703313836300omplex(aTP_Lamm_jk(Uu2),Uua),Uub),Uuc) = groups1794756597179926696omplex(aa_com2862003112617944841omplex(aTP_Lamm_jj(Uua),Uuc),collect_VEBT_VEBT(aa_com3651963455866140051VEBT_o(aa_fun7268111043052431330VEBT_o(aTP_Lamm_fx(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_443
tff(fact_9542_ATP_Olambda__444,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_jg(Uu2),Uua),Uub),Uuc) = groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_je(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_jf(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_444
tff(fact_9543_ATP_Olambda__445,axiom,
    ! [Uu2: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_jb(Uu2),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_iz(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_ja(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_445
tff(fact_9544_ATP_Olambda__446,axiom,
    ! [Uu2: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_iw(Uu2),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_iu(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_iv(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_446
tff(fact_9545_ATP_Olambda__447,axiom,
    ! [Uu2: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_ir(Uu2),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_iq(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_fu(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_447
tff(fact_9546_ATP_Olambda__448,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE6480751511521678896_nat_o,Uuc: nat] : ( aa_nat_nat(aa_fun6249118799011721995at_nat(aa_fun2705500642357200114at_nat(aTP_Lamm_in(Uu2),Uua),Uub),Uuc) = groups771621172384141258BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_im(Uua),Uuc),collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_ft(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_448
tff(fact_9547_ATP_Olambda__449,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_wd(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_wc(Uu2),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ) ).

% ATP.lambda_449
tff(fact_9548_ATP_Olambda__450,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : ( aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yt(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ) ).

% ATP.lambda_450
tff(fact_9549_ATP_Olambda__451,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : ( aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yr(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ) ).

% ATP.lambda_451
tff(fact_9550_ATP_Olambda__452,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : ( aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yp(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ) ).

% ATP.lambda_452
tff(fact_9551_ATP_Olambda__453,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : ( aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_yq(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ) ).

% ATP.lambda_453
tff(fact_9552_ATP_Olambda__454,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : ( aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_qk(Uu2),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(Uu2),Uuc)) ) ).

% ATP.lambda_454
tff(fact_9553_ATP_Olambda__455,axiom,
    ! [Uu2: uint32,Uua: nat,Uub: uint32,Uuc: nat] : ( aa_nat_uint32(aa_uin238090125092722531uint32(aa_nat7697544214703702448uint32(aTP_Lamm_ql(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(power_power_uint32(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_uint32(power_power_uint32(Uu2),Uuc)) ) ).

% ATP.lambda_455
tff(fact_9554_ATP_Olambda__456,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : ( aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_qn(Uu2),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,Uu2),Uuc)) ) ).

% ATP.lambda_456
tff(fact_9555_ATP_Olambda__457,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : ( aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_qm(Uu2),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(Uu2),Uuc)) ) ).

% ATP.lambda_457
tff(fact_9556_ATP_Olambda__458,axiom,
    ! [Uu2: fun_VE3165209603783583043T_VEBT,Uua: set_VEBT_VEBT,Uub: fun_VEBT_VEBT_o,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aa_set7749190676507875235VEBT_o(aTP_Lamm_gn(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),image_3375948659692109573T_VEBT(Uu2,Uua))
        & aa_VEBT_VEBT_o(Uub,Uuc) ) ) ).

% ATP.lambda_458
tff(fact_9557_ATP_Olambda__459,axiom,
    ! [Uu2: fun_VEBT_VEBT_real,Uua: set_VEBT_VEBT,Uub: fun_real_o,Uuc: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aa_set5216747956267341787real_o(aTP_Lamm_gr(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),image_VEBT_VEBT_real(Uu2,Uua))
        & aa_real_o(Uub,Uuc) ) ) ).

% ATP.lambda_459
tff(fact_9558_ATP_Olambda__460,axiom,
    ! [Uu2: fun_co4340422964966740447T_VEBT,Uua: set_complex,Uub: fun_VEBT_VEBT_o,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aa_set5809115984592220781VEBT_o(aTP_Lamm_gv(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),image_932796090930683071T_VEBT(Uu2,Uua))
        & aa_VEBT_VEBT_o(Uub,Uuc) ) ) ).

% ATP.lambda_460
tff(fact_9559_ATP_Olambda__461,axiom,
    ! [Uu2: fun_complex_real,Uua: set_complex,Uub: fun_real_o,Uuc: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aa_set8509023117188218277real_o(aTP_Lamm_gx(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),image_complex_real(Uu2,Uua))
        & aa_real_o(Uub,Uuc) ) ) ).

% ATP.lambda_461
tff(fact_9560_ATP_Olambda__462,axiom,
    ! [Uu2: fun_real_VEBT_VEBT,Uua: set_real,Uub: fun_VEBT_VEBT_o,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aa_set6460758714441890155VEBT_o(aTP_Lamm_gp(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),image_real_VEBT_VEBT(Uu2,Uua))
        & aa_VEBT_VEBT_o(Uub,Uuc) ) ) ).

% ATP.lambda_462
tff(fact_9561_ATP_Olambda__463,axiom,
    ! [Uu2: fun_real_real,Uua: set_real,Uub: fun_real_o,Uuc: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aa_set1691088162573342115real_o(aTP_Lamm_gt(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),image_real_real(Uu2,Uua))
        & aa_real_o(Uub,Uuc) ) ) ).

% ATP.lambda_463
tff(fact_9562_ATP_Olambda__464,axiom,
    ! [Uu2: fun_nat_VEBT_VEBT,Uua: set_nat,Uub: fun_VEBT_VEBT_o,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aa_set2181257101989441679VEBT_o(aTP_Lamm_gz(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),image_nat_VEBT_VEBT(Uu2,Uua))
        & aa_VEBT_VEBT_o(Uub,Uuc) ) ) ).

% ATP.lambda_464
tff(fact_9563_ATP_Olambda__465,axiom,
    ! [Uu2: fun_nat_real,Uua: set_nat,Uub: fun_real_o,Uuc: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aa_set8653532325685533383real_o(aTP_Lamm_hb(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),image_nat_real(Uu2,Uua))
        & aa_real_o(Uub,Uuc) ) ) ).

% ATP.lambda_465
tff(fact_9564_ATP_Olambda__466,axiom,
    ! [Uu2: fun_int_VEBT_VEBT,Uua: set_int,Uub: fun_VEBT_VEBT_o,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aa_set2625704858979348715VEBT_o(aTP_Lamm_hd(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),image_int_VEBT_VEBT(Uu2,Uua))
        & aa_VEBT_VEBT_o(Uub,Uuc) ) ) ).

% ATP.lambda_466
tff(fact_9565_ATP_Olambda__467,axiom,
    ! [Uu2: fun_int_real,Uua: set_int,Uub: fun_real_o,Uuc: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aa_set7127480517943840035real_o(aTP_Lamm_hf(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),image_int_real(Uu2,Uua))
        & aa_real_o(Uub,Uuc) ) ) ).

% ATP.lambda_467
tff(fact_9566_ATP_Olambda__468,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE2962686792283380846VEBT_o,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(aa_fun8771395773580142140VEBT_o(aTP_Lamm_os(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_468
tff(fact_9567_ATP_Olambda__469,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_li8107907266037299598VEBT_o,Uub: literal,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_lit1284886718724587071VEBT_o(aa_fun839249340096772540VEBT_o(aTP_Lamm_pk(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_VEBT_VEBT_o(aa_lit1284886718724587071VEBT_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_469
tff(fact_9568_ATP_Olambda__470,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_re7829509182167823952VEBT_o,Uub: real,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_rea6225614514526338833VEBT_o(aa_fun2719675890530006366VEBT_o(aTP_Lamm_oy(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_VEBT_VEBT_o(aa_rea6225614514526338833VEBT_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_470
tff(fact_9569_ATP_Olambda__471,axiom,
    ! [Uu2: set_complex,Uua: fun_VE7631951351884037842plex_o,Uub: vEBT_VEBT,Uuc: complex] :
      ( aa_complex_o(aa_VEB8036846806599048395plex_o(aa_fun5896829868240219618plex_o(aTP_Lamm_jh(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & aa_complex_o(aa_VEB8036846806599048395plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_471
tff(fact_9570_ATP_Olambda__472,axiom,
    ! [Uu2: set_complex,Uua: fun_Co8975588572093281791plex_o,Uub: code_integer,Uuc: complex] :
      ( aa_complex_o(aa_Cod7908732550325689320plex_o(aa_fun2498529521300939132plex_o(aTP_Lamm_jz(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & aa_complex_o(aa_Cod7908732550325689320plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_472
tff(fact_9571_ATP_Olambda__473,axiom,
    ! [Uu2: set_complex,Uua: fun_re7319081385389557872plex_o,Uub: real,Uuc: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_jl(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_473
tff(fact_9572_ATP_Olambda__474,axiom,
    ! [Uu2: set_complex,Uua: fun_na1554462459066888140plex_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_jp(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_474
tff(fact_9573_ATP_Olambda__475,axiom,
    ! [Uu2: set_complex,Uua: fun_in5065666251602127472plex_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_ju(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_475
tff(fact_9574_ATP_Olambda__476,axiom,
    ! [Uu2: set_literal,Uua: fun_VE8371440389858774350eral_o,Uub: vEBT_VEBT,Uuc: literal] :
      ( aa_literal_o(aa_VEB1719478621182999551eral_o(aa_fun5643091140434266556eral_o(aTP_Lamm_pe(Uu2),Uua),Uub),Uuc)
    <=> ( member_literal(Uuc,Uu2)
        & aa_literal_o(aa_VEB1719478621182999551eral_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_476
tff(fact_9575_ATP_Olambda__477,axiom,
    ! [Uu2: set_literal,Uua: fun_li4177005666941401710eral_o,Uub: literal,Uuc: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(aa_fun6827806182750346556eral_o(aTP_Lamm_pq(Uu2),Uua),Uub),Uuc)
    <=> ( member_literal(Uuc,Uu2)
        & aa_literal_o(aa_lit5986169069484365087eral_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_477
tff(fact_9576_ATP_Olambda__478,axiom,
    ! [Uu2: set_literal,Uua: fun_re3850508403959675952eral_o,Uub: real,Uuc: literal] :
      ( aa_literal_o(aa_rea1838623199774720753eral_o(aa_fun1820749043261144222eral_o(aTP_Lamm_ph(Uu2),Uua),Uub),Uuc)
    <=> ( member_literal(Uuc,Uu2)
        & aa_literal_o(aa_rea1838623199774720753eral_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_478
tff(fact_9577_ATP_Olambda__479,axiom,
    ! [Uu2: set_real,Uua: fun_VE3823611782367660756real_o,Uub: vEBT_VEBT,Uuc: real] :
      ( aa_real_o(aa_VEB641519613517496269real_o(aa_fun5792667679087400038real_o(aTP_Lamm_ov(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_real_o(aa_VEB641519613517496269real_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_479
tff(fact_9578_ATP_Olambda__480,axiom,
    ! [Uu2: set_real,Uua: fun_li5129792558484764084real_o,Uub: literal,Uuc: real] :
      ( aa_real_o(aa_lit1368651074454191789real_o(aa_fun6987840782327099302real_o(aTP_Lamm_pn(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_real_o(aa_lit1368651074454191789real_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_480
tff(fact_9579_ATP_Olambda__481,axiom,
    ! [Uu2: set_real,Uua: fun_real_fun_real_o,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_fun4467907209059791804real_o(aTP_Lamm_pb(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_real_o(aa_real_fun_real_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_481
tff(fact_9580_ATP_Olambda__482,axiom,
    ! [Uu2: set_nat,Uua: fun_VE6480751511521678896_nat_o,Uub: vEBT_VEBT,Uuc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(aa_fun1998352179508271774_nat_o(aTP_Lamm_ik(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & aa_nat_o(aa_VEB2314620471664397097_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_482
tff(fact_9581_ATP_Olambda__483,axiom,
    ! [Uu2: set_nat,Uua: fun_Co1375048237525739357_nat_o,Uub: code_integer,Uuc: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_jc(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_483
tff(fact_9582_ATP_Olambda__484,axiom,
    ! [Uu2: set_nat,Uua: fun_co5676208753217392844_nat_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ix(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_484
tff(fact_9583_ATP_Olambda__485,axiom,
    ! [Uu2: set_nat,Uua: fun_real_fun_nat_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_io(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_485
tff(fact_9584_ATP_Olambda__486,axiom,
    ! [Uu2: set_nat,Uua: fun_int_fun_nat_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_is(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_486
tff(fact_9585_ATP_Olambda__487,axiom,
    ! [Uu2: set_int,Uua: fun_VE4235606543851232980_int_o,Uub: vEBT_VEBT,Uuc: int] :
      ( aa_int_o(aa_VEB493578126231709133_int_o(aa_fun8464664609713939430_int_o(aTP_Lamm_ps(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & aa_int_o(aa_VEB493578126231709133_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_487
tff(fact_9586_ATP_Olambda__488,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE2962686792283380846VEBT_o,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(aa_fun8771395773580142140VEBT_o(aTP_Lamm_or(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_488
tff(fact_9587_ATP_Olambda__489,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE1909044406494083811eger_o,Uub: code_integer,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_Cod5040069908416755522VEBT_o(aa_fun1637734041313256866VEBT_o(aTP_Lamm_fz(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_Code_integer_o(aa_VEB1409644639000594780eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_489
tff(fact_9588_ATP_Olambda__490,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE7631951351884037842plex_o,Uub: complex,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_com3651963455866140051VEBT_o(aa_fun7268111043052431330VEBT_o(aTP_Lamm_fx(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_complex_o(aa_VEB8036846806599048395plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_490
tff(fact_9589_ATP_Olambda__491,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE8371440389858774350eral_o,Uub: literal,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_lit1284886718724587071VEBT_o(aa_fun5379558016612791804VEBT_o(aTP_Lamm_pd(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_literal_o(aa_VEB1719478621182999551eral_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_491
tff(fact_9590_ATP_Olambda__492,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE3823611782367660756real_o,Uub: real,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_rea6225614514526338833VEBT_o(aa_fun575193042032787426VEBT_o(aTP_Lamm_ou(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_real_o(aa_VEB641519613517496269real_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_492
tff(fact_9591_ATP_Olambda__493,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE6480751511521678896_nat_o,Uub: nat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_ft(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_nat_o(aa_VEB2314620471664397097_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_493
tff(fact_9592_ATP_Olambda__494,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE4235606543851232980_int_o,Uub: int,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_int8839263053405020433VEBT_o(aa_fun8807180201667926370VEBT_o(aTP_Lamm_fv(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & aa_int_o(aa_VEB493578126231709133_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_494
tff(fact_9593_ATP_Olambda__495,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Co8975588572093281791plex_o,Uub: complex,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_com4877492900066995176eger_o(aa_fun3025727905316630780eger_o(aTP_Lamm_kc(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu2)
        & aa_complex_o(aa_Cod7908732550325689320plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_495
tff(fact_9594_ATP_Olambda__496,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Co1375048237525739357_nat_o,Uub: nat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_jf(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu2)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_496
tff(fact_9595_ATP_Olambda__497,axiom,
    ! [Uu2: set_complex,Uua: fun_co5676208753217392844_nat_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_ja(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_497
tff(fact_9596_ATP_Olambda__498,axiom,
    ! [Uu2: set_literal,Uua: fun_li8107907266037299598VEBT_o,Uub: vEBT_VEBT,Uuc: literal] :
      ( aa_literal_o(aa_VEB1719478621182999551eral_o(aa_fun1102782463918247292eral_o(aTP_Lamm_pj(Uu2),Uua),Uub),Uuc)
    <=> ( member_literal(Uuc,Uu2)
        & aa_VEBT_VEBT_o(aa_lit1284886718724587071VEBT_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_498
tff(fact_9597_ATP_Olambda__499,axiom,
    ! [Uu2: set_literal,Uua: fun_li4177005666941401710eral_o,Uub: literal,Uuc: literal] :
      ( aa_literal_o(aa_lit5986169069484365087eral_o(aa_fun6827806182750346556eral_o(aTP_Lamm_pp(Uu2),Uua),Uub),Uuc)
    <=> ( member_literal(Uuc,Uu2)
        & aa_literal_o(aa_lit5986169069484365087eral_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_499
tff(fact_9598_ATP_Olambda__500,axiom,
    ! [Uu2: set_literal,Uua: fun_li5129792558484764084real_o,Uub: real,Uuc: literal] :
      ( aa_literal_o(aa_rea1838623199774720753eral_o(aa_fun5708556627802011170eral_o(aTP_Lamm_pm(Uu2),Uua),Uub),Uuc)
    <=> ( member_literal(Uuc,Uu2)
        & aa_real_o(aa_lit1368651074454191789real_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_500
tff(fact_9599_ATP_Olambda__501,axiom,
    ! [Uu2: set_real,Uua: fun_re7829509182167823952VEBT_o,Uub: vEBT_VEBT,Uuc: real] :
      ( aa_real_o(aa_VEB641519613517496269real_o(aa_fun7937150527584618978real_o(aTP_Lamm_ox(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_VEBT_VEBT_o(aa_rea6225614514526338833VEBT_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_501
tff(fact_9600_ATP_Olambda__502,axiom,
    ! [Uu2: set_real,Uua: fun_re3424108668441675905eger_o,Uub: code_integer,Uuc: real] :
      ( aa_real_o(aa_Cod1315720040876898922real_o(aa_fun4247632250071955964real_o(aTP_Lamm_ga(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_Code_integer_o(aa_rea4090721512283939562eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_502
tff(fact_9601_ATP_Olambda__503,axiom,
    ! [Uu2: set_real,Uua: fun_re7319081385389557872plex_o,Uub: complex,Uuc: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_fy(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_503
tff(fact_9602_ATP_Olambda__504,axiom,
    ! [Uu2: set_real,Uua: fun_re3850508403959675952eral_o,Uub: literal,Uuc: real] :
      ( aa_real_o(aa_lit1368651074454191789real_o(aa_fun3100033197786232354real_o(aTP_Lamm_pg(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_literal_o(aa_rea1838623199774720753eral_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_504
tff(fact_9603_ATP_Olambda__505,axiom,
    ! [Uu2: set_real,Uua: fun_real_fun_real_o,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_fun4467907209059791804real_o(aTP_Lamm_pa(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_real_o(aa_real_fun_real_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_505
tff(fact_9604_ATP_Olambda__506,axiom,
    ! [Uu2: set_real,Uua: fun_real_fun_nat_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_fu(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_506
tff(fact_9605_ATP_Olambda__507,axiom,
    ! [Uu2: set_real,Uua: fun_real_fun_int_o,Uub: int,Uuc: real] :
      ( aa_real_o(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_fw(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & aa_int_o(aa_real_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_507
tff(fact_9606_ATP_Olambda__508,axiom,
    ! [Uu2: set_nat,Uua: fun_na1554462459066888140plex_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_js(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_508
tff(fact_9607_ATP_Olambda__509,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_gb(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_509
tff(fact_9608_ATP_Olambda__510,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_fun_int_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_gc(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_510
tff(fact_9609_ATP_Olambda__511,axiom,
    ! [Uu2: set_int,Uua: fun_in5065666251602127472plex_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_jx(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_511
tff(fact_9610_ATP_Olambda__512,axiom,
    ! [Uu2: set_int,Uua: fun_int_fun_nat_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_iv(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_512
tff(fact_9611_ATP_Olambda__513,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE9219535225822561060nteger,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(aa_fun8260802627938895110VEBT_o(aTP_Lamm_hs(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_VEB5935388676523804373nteger(Uua,Uuc) = aa_VEB5935388676523804373nteger(Uua,Uub) ) ) ) ).

% ATP.lambda_513
tff(fact_9612_ATP_Olambda__514,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(aa_fun7842805400471075189VEBT_o(aTP_Lamm_hq(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_VEBT_VEBT_complex(Uua,Uuc) = aa_VEBT_VEBT_complex(Uua,Uub) ) ) ) ).

% ATP.lambda_514
tff(fact_9613_ATP_Olambda__515,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(aa_fun8512903131112361427VEBT_o(aTP_Lamm_hm(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_VEBT_VEBT_nat(Uua,Uuc) = aa_VEBT_VEBT_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_515
tff(fact_9614_ATP_Olambda__516,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_int,Uub: vEBT_VEBT,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(aa_fun211421702454883447VEBT_o(aTP_Lamm_ho(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_VEBT_VEBT_int(Uua,Uuc) = aa_VEBT_VEBT_int(Uua,Uub) ) ) ) ).

% ATP.lambda_516
tff(fact_9615_ATP_Olambda__517,axiom,
    ! [Uu2: set_real,Uua: fun_re6978948838062957830nteger,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_fun7085772160498067374real_o(aTP_Lamm_ht(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_real_Code_integer(Uua,Uuc) = aa_real_Code_integer(Uua,Uub) ) ) ) ).

% ATP.lambda_517
tff(fact_9616_ATP_Olambda__518,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_fun3021595049966113949real_o(aTP_Lamm_hr(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_real_complex(Uua,Uuc) = aa_real_complex(Uua,Uub) ) ) ) ).

% ATP.lambda_518
tff(fact_9617_ATP_Olambda__519,axiom,
    ! [Uu2: set_real,Uua: fun_real_nat,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_fun8229752488527876603real_o(aTP_Lamm_hn(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_real_nat(Uua,Uuc) = aa_real_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_519
tff(fact_9618_ATP_Olambda__520,axiom,
    ! [Uu2: set_real,Uua: fun_real_int,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_fun7132669507883878047real_o(aTP_Lamm_hp(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_real_int(Uua,Uuc) = aa_real_int(Uua,Uub) ) ) ) ).

% ATP.lambda_520
tff(fact_9619_ATP_Olambda__521,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_hu(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_nat_nat(Uua,Uuc) = aa_nat_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_521
tff(fact_9620_ATP_Olambda__522,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_int,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun4569312790843823667_nat_o(aTP_Lamm_hv(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_nat_int(Uua,Uuc) = aa_nat_int(Uua,Uub) ) ) ) ).

% ATP.lambda_522
tff(fact_9621_ATP_Olambda__523,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE9219535225822561060nteger,Uub: code_integer,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_Cod5040069908416755522VEBT_o(aa_fun6317896443157543913VEBT_o(aTP_Lamm_rj(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_VEB5935388676523804373nteger(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_523
tff(fact_9622_ATP_Olambda__524,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: complex,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_com3651963455866140051VEBT_o(aa_fun4726854402537413643VEBT_o(aTP_Lamm_od(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_VEBT_VEBT_complex(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_524
tff(fact_9623_ATP_Olambda__525,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: nat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(aa_fun8313996525498199887VEBT_o(aTP_Lamm_nt(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_VEBT_VEBT_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_525
tff(fact_9624_ATP_Olambda__526,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Co1900716531650368264omplex,Uub: complex,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_com4877492900066995176eger_o(aa_fun6433599355428082873eger_o(aTP_Lamm_ol(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu2)
        & ( aa_Cod5621350732091595593omplex(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_526
tff(fact_9625_ATP_Olambda__527,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Code_integer_nat,Uub: nat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aa_fun4176659983681010805eger_o(aTP_Lamm_ob(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu2)
        & ( aa_Code_integer_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_527
tff(fact_9626_ATP_Olambda__528,axiom,
    ! [Uu2: set_complex,Uua: fun_co8153928105989009288nteger,Uub: code_integer,Uuc: complex] :
      ( aa_complex_o(aa_Cod7908732550325689320plex_o(aa_fun8663194649151195321plex_o(aTP_Lamm_sg(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & ( aa_com4217951221296830281nteger(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_528
tff(fact_9627_ATP_Olambda__529,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_nat,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun3931428123345706963plex_o(aTP_Lamm_nz(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & ( aa_complex_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_529
tff(fact_9628_ATP_Olambda__530,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_int,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun1485481922755881755plex_o(aTP_Lamm_si(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & ( aa_complex_int(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_530
tff(fact_9629_ATP_Olambda__531,axiom,
    ! [Uu2: set_literal,Uua: fun_li486744618827423812nteger,Uub: code_integer,Uuc: literal] :
      ( aa_literal_o(aa_Cod3435528066990957986eral_o(aa_fun5342429524804797673eral_o(aTP_Lamm_rp(Uu2),Uua),Uub),Uuc)
    <=> ( member_literal(Uuc,Uu2)
        & ( aa_lit3353659836003880949nteger(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_531
tff(fact_9630_ATP_Olambda__532,axiom,
    ! [Uu2: set_literal,Uua: fun_literal_complex,Uub: complex,Uuc: literal] :
      ( aa_literal_o(aa_com6030452599096385139eral_o(aa_fun1482976424769362443eral_o(aTP_Lamm_rg(Uu2),Uua),Uub),Uuc)
    <=> ( member_literal(Uuc,Uu2)
        & ( aa_literal_complex(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_532
tff(fact_9631_ATP_Olambda__533,axiom,
    ! [Uu2: set_literal,Uua: fun_literal_nat,Uub: nat,Uuc: literal] :
      ( aa_literal_o(aa_nat_fun_literal_o(aa_fun4198428272352532303eral_o(aTP_Lamm_qz(Uu2),Uua),Uub),Uuc)
    <=> ( member_literal(Uuc,Uu2)
        & ( aa_literal_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_533
tff(fact_9632_ATP_Olambda__534,axiom,
    ! [Uu2: set_real,Uua: fun_re6978948838062957830nteger,Uub: code_integer,Uuc: real] :
      ( aa_real_o(aa_Cod1315720040876898922real_o(aa_fun4638768144124725693real_o(aTP_Lamm_rm(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_real_Code_integer(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_534
tff(fact_9633_ATP_Olambda__535,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: complex,Uuc: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aa_fun8514437842610354715real_o(aTP_Lamm_of(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_real_complex(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_535
tff(fact_9634_ATP_Olambda__536,axiom,
    ! [Uu2: set_real,Uua: fun_real_nat,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun6618657382008502231real_o(aTP_Lamm_nv(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_real_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_536
tff(fact_9635_ATP_Olambda__537,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_Code_integer,Uub: code_integer,Uuc: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aa_fun262808356775936629_nat_o(aTP_Lamm_sb(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_nat_Code_integer(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_537
tff(fact_9636_ATP_Olambda__538,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_complex,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun8139692174331916755_nat_o(aTP_Lamm_oh(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_nat_complex(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_538
tff(fact_9637_ATP_Olambda__539,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_rs(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_nat_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_539
tff(fact_9638_ATP_Olambda__540,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_int,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_sd(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_nat_int(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_540
tff(fact_9639_ATP_Olambda__541,axiom,
    ! [Uu2: set_int,Uua: fun_int_complex,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun3347742875685192987_int_o(aTP_Lamm_oj(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & ( aa_int_complex(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_541
tff(fact_9640_ATP_Olambda__542,axiom,
    ! [Uu2: set_int,Uua: fun_int_nat,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_nx(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & ( aa_int_nat(Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_542
tff(fact_9641_ATP_Olambda__543,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex,Uuc: nat] : ( aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_qg(Uu2),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu2),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ) ).

% ATP.lambda_543
tff(fact_9642_ATP_Olambda__544,axiom,
    ! [Uu2: uint32,Uua: nat,Uub: uint32,Uuc: nat] : ( aa_nat_uint32(aa_uin238090125092722531uint32(aa_nat7697544214703702448uint32(aTP_Lamm_qh(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(power_power_uint32(Uu2),Uuc)),aa_nat_uint32(power_power_uint32(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ) ).

% ATP.lambda_544
tff(fact_9643_ATP_Olambda__545,axiom,
    ! [Uu2: real,Uua: nat,Uub: real,Uuc: nat] : ( aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_qj(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),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_545
tff(fact_9644_ATP_Olambda__546,axiom,
    ! [Uu2: int,Uua: nat,Uub: int,Uuc: nat] : ( aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_qi(Uu2),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu2),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ) ).

% ATP.lambda_546
tff(fact_9645_ATP_Olambda__547,axiom,
    ! [Uu2: fun_VE3165209603783583043T_VEBT,Uua: set_VEBT_VEBT,Uub: fun_VEBT_VEBT_o,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aa_set7749190676507875235VEBT_o(aTP_Lamm_go(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uua)
        & aa_VEBT_VEBT_o(Uub,aa_VEB5832953974749077948T_VEBT(Uu2,Uuc)) ) ) ).

% ATP.lambda_547
tff(fact_9646_ATP_Olambda__548,axiom,
    ! [Uu2: fun_VEBT_VEBT_real,Uua: set_VEBT_VEBT,Uub: fun_real_o,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun7216158613899282224VEBT_o(aa_set4022976891253242279VEBT_o(aTP_Lamm_gs(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uua)
        & aa_real_o(Uub,aa_VEBT_VEBT_real(Uu2,Uuc)) ) ) ).

% ATP.lambda_548
tff(fact_9647_ATP_Olambda__549,axiom,
    ! [Uu2: fun_co4340422964966740447T_VEBT,Uua: set_complex,Uub: fun_VEBT_VEBT_o,Uuc: complex] :
      ( aa_complex_o(aa_fun7745313613784576798plex_o(aa_set1943973126016318227plex_o(aTP_Lamm_gw(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uua)
        & aa_VEBT_VEBT_o(Uub,aa_complex_VEBT_VEBT(Uu2,Uuc)) ) ) ).

% ATP.lambda_549
tff(fact_9648_ATP_Olambda__550,axiom,
    ! [Uu2: fun_complex_real,Uua: set_complex,Uub: fun_real_o,Uuc: complex] :
      ( aa_complex_o(aa_fun3391577472191272058plex_o(aa_set4195536971368458275plex_o(aTP_Lamm_gy(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uua)
        & aa_real_o(Uub,aa_complex_real(Uu2,Uuc)) ) ) ).

% ATP.lambda_550
tff(fact_9649_ATP_Olambda__551,axiom,
    ! [Uu2: fun_real_VEBT_VEBT,Uua: set_real,Uub: fun_VEBT_VEBT_o,Uuc: real] :
      ( aa_real_o(aa_fun2752195911241646752real_o(aa_set5716891373128775959real_o(aTP_Lamm_gq(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uua)
        & aa_VEBT_VEBT_o(Uub,aa_real_VEBT_VEBT(Uu2,Uuc)) ) ) ).

% ATP.lambda_551
tff(fact_9650_ATP_Olambda__552,axiom,
    ! [Uu2: fun_real_real,Uua: set_real,Uub: fun_real_o,Uuc: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aa_set1691088162573342115real_o(aTP_Lamm_gu(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uua)
        & aa_real_o(Uub,aa_real_real(Uu2,Uuc)) ) ) ).

% ATP.lambda_552
tff(fact_9651_ATP_Olambda__553,axiom,
    ! [Uu2: fun_nat_VEBT_VEBT,Uua: set_nat,Uub: fun_VEBT_VEBT_o,Uuc: nat] :
      ( aa_nat_o(aa_fun370934926479296380_nat_o(aa_set5037496533744297423_nat_o(aTP_Lamm_ha(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uua)
        & aa_VEBT_VEBT_o(Uub,aa_nat_VEBT_VEBT(Uu2,Uuc)) ) ) ).

% ATP.lambda_553
tff(fact_9652_ATP_Olambda__554,axiom,
    ! [Uu2: fun_nat_real,Uua: set_nat,Uub: fun_real_o,Uuc: nat] :
      ( aa_nat_o(aa_fun966921962273399000_nat_o(aa_set47184369695174563_nat_o(aTP_Lamm_hc(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uua)
        & aa_real_o(Uub,aa_nat_real(Uu2,Uuc)) ) ) ).

% ATP.lambda_554
tff(fact_9653_ATP_Olambda__555,axiom,
    ! [Uu2: fun_int_VEBT_VEBT,Uua: set_int,Uub: fun_VEBT_VEBT_o,Uuc: int] :
      ( aa_int_o(aa_fun7773264617901384224_int_o(aa_set2765448116205184791_int_o(aTP_Lamm_he(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uua)
        & aa_VEBT_VEBT_o(Uub,aa_int_VEBT_VEBT(Uu2,Uuc)) ) ) ).

% ATP.lambda_555
tff(fact_9654_ATP_Olambda__556,axiom,
    ! [Uu2: fun_int_real,Uua: set_int,Uub: fun_real_o,Uuc: int] :
      ( aa_int_o(aa_fun8369251653695486844_int_o(aa_set5431007883013265571_int_o(aTP_Lamm_hg(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uua)
        & aa_real_o(Uub,aa_int_real(Uu2,Uuc)) ) ) ).

% ATP.lambda_556
tff(fact_9655_ATP_Olambda__557,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_assn,Uub: fun_VEBT_VEBT_assn,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5142337294461892954VEBT_o(aa_fun876682331325704374VEBT_o(aTP_Lamm_mf(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBT_assn(Uua,Uuc)),aa_VEBT_VEBT_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_557
tff(fact_9656_ATP_Olambda__558,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5889122065779655690VEBT_o(aa_fun1978406156034293430VEBT_o(aTP_Lamm_lt(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_VEBT_VEBT_rat(Uua,Uuc)),aa_VEBT_VEBT_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_558
tff(fact_9657_ATP_Olambda__559,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun6395433285121891940eger_o(aa_fun4095556841599367307eger_o(aTP_Lamm_md(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Code_integer_rat(Uua,Uuc)),aa_Code_integer_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_559
tff(fact_9658_ATP_Olambda__560,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      ( aa_complex_o(aa_fun5099116602557847362plex_o(aa_fun4586712283274990936plex_o(aTP_Lamm_mb(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_560
tff(fact_9659_ATP_Olambda__561,axiom,
    ! [Uu2: set_real,Uua: fun_real_assn,Uub: fun_real_assn,Uuc: real] :
      ( aa_real_o(aa_fun7580339382299851254real_o(aa_fun4959323011771448382real_o(aTP_Lamm_mh(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_real_assn(Uua,Uuc)),aa_real_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_561
tff(fact_9660_ATP_Olambda__562,axiom,
    ! [Uu2: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_lv(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_562
tff(fact_9661_ATP_Olambda__563,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_assn,Uub: fun_nat_assn,Uuc: nat] :
      ( aa_nat_o(aa_fun7183233715851567790_nat_o(aa_fun8595640264602358482_nat_o(aTP_Lamm_mj(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(Uua,Uuc)),aa_nat_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_563
tff(fact_9662_ATP_Olambda__564,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_lx(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_564
tff(fact_9663_ATP_Olambda__565,axiom,
    ! [Uu2: set_int,Uua: fun_int_assn,Uub: fun_int_assn,Uuc: int] :
      ( aa_int_o(aa_fun202081646129719286_int_o(aa_fun871127997271750846_int_o(aTP_Lamm_ml(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(Uua,Uuc)),aa_int_assn(Uub,Uuc)) != one_one_assn ) ) ) ).

% ATP.lambda_565
tff(fact_9664_ATP_Olambda__566,axiom,
    ! [Uu2: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_lz(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_566
tff(fact_9665_ATP_Olambda__567,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: fun_VEBT_VEBT_uint32,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun6272599145770227614VEBT_o(aa_fun1156574460560677430VEBT_o(aTP_Lamm_kz(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_VEBT_VEBT_uint32(Uua,Uuc)),aa_VEBT_VEBT_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_567
tff(fact_9666_ATP_Olambda__568,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_ll(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_568
tff(fact_9667_ATP_Olambda__569,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Co1554742298434871494uint32,Uub: fun_Co1554742298434871494uint32,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun8926496376162642256eger_o(aa_fun1119956648780089443eger_o(aTP_Lamm_lj(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu2)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_Cod8401701657498852871uint32(Uua,Uuc)),aa_Cod8401701657498852871uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_569
tff(fact_9668_ATP_Olambda__570,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_uint32,Uub: fun_complex_uint32,Uuc: complex] :
      ( aa_complex_o(aa_fun436548534304416686plex_o(aa_fun2751405167760845232plex_o(aTP_Lamm_lh(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu2)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_complex_uint32(Uua,Uuc)),aa_complex_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_570
tff(fact_9669_ATP_Olambda__571,axiom,
    ! [Uu2: set_real,Uua: fun_real_uint32,Uub: fun_real_uint32,Uuc: real] :
      ( aa_real_o(aa_fun7556296611544133298real_o(aa_fun5572330763077102134real_o(aTP_Lamm_lb(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_real_uint32(Uua,Uuc)),aa_real_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_571
tff(fact_9670_ATP_Olambda__572,axiom,
    ! [Uu2: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ln(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_572
tff(fact_9671_ATP_Olambda__573,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_uint32,Uub: fun_nat_uint32,Uuc: nat] :
      ( aa_nat_o(aa_fun1182053767699555178_nat_o(aa_fun6957237400125433034_nat_o(aTP_Lamm_ld(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_nat_uint32(Uua,Uuc)),aa_nat_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_573
tff(fact_9672_ATP_Olambda__574,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_lp(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_574
tff(fact_9673_ATP_Olambda__575,axiom,
    ! [Uu2: set_int,Uua: fun_int_uint32,Uub: fun_int_uint32,Uuc: int] :
      ( aa_int_o(aa_fun6992502331602369202_int_o(aa_fun3369613587901038774_int_o(aTP_Lamm_lf(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & ( aa_uint32_uint32(plus_plus_uint32(aa_int_uint32(Uua,Uuc)),aa_int_uint32(Uub,Uuc)) != zero_zero_uint32 ) ) ) ).

% ATP.lambda_575
tff(fact_9674_ATP_Olambda__576,axiom,
    ! [Uu2: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_lr(Uu2),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu2)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_576
tff(fact_9675_ATP_Olambda__577,axiom,
    ! [Uu2: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(aa_set5977240410424667023T_assn(aTP_Lamm_wj(Uu2),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uuc),Uub)),
          aa_o_assn(pure_assn,
            ( ( Uua = vEBT_set_vebt(Uuc) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Uuc),Uu2) ))) ) ).

% ATP.lambda_577
tff(fact_9676_ATP_Olambda__578,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: $o] : ( aa_o_assn(aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_dq(Uu2),Uua),Uub),(Uuc)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu2),Uua)),aa_o_assn(pure_assn,(Uuc) = aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu2),Uub))) ) ).

% ATP.lambda_578
tff(fact_9677_ATP_Olambda__579,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : ( aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_dr(Uu2),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu2),Uua)),aa_o_assn(pure_assn,Uuc = vEBT_vebt_succ(Uu2,Uub))) ) ).

% ATP.lambda_579
tff(fact_9678_ATP_Olambda__580,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : ( aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_ds(Uu2),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu2),Uua)),aa_o_assn(pure_assn,Uuc = vEBT_vebt_pred(Uu2,Uub))) ) ).

% ATP.lambda_580
tff(fact_9679_ATP_Olambda__581,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_wc(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ) ).

% ATP.lambda_581
tff(fact_9680_ATP_Olambda__582,axiom,
    ! [Uu2: set_complex,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co8975588572093281791plex_o,Uuc: code_integer] : ( aa_Cod5621350732091595593omplex(aa_fun9008897831946212043omplex(aa_fun8319717443755388649omplex(aTP_Lamm_ka(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_Cod3397930797123873268omplex(Uua,Uuc),collect_complex(aa_Cod7908732550325689320plex_o(aa_fun2498529521300939132plex_o(aTP_Lamm_jz(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_582
tff(fact_9681_ATP_Olambda__583,axiom,
    ! [Uu2: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: int] : ( aa_int_complex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_jv(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_ju(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_583
tff(fact_9682_ATP_Olambda__584,axiom,
    ! [Uu2: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : ( aa_nat_complex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_jq(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_jp(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_584
tff(fact_9683_ATP_Olambda__585,axiom,
    ! [Uu2: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: real] : ( aa_real_complex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_jm(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_jl(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_585
tff(fact_9684_ATP_Olambda__586,axiom,
    ! [Uu2: set_complex,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE7631951351884037842plex_o,Uuc: vEBT_VEBT] : ( aa_VEBT_VEBT_complex(aa_fun1953711888481404901omplex(aa_fun5829985485761010584omplex(aTP_Lamm_ji(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_VEB5608662635949521omplex(Uua,Uuc),collect_complex(aa_VEB8036846806599048395plex_o(aa_fun5896829868240219618plex_o(aTP_Lamm_jh(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_586
tff(fact_9685_ATP_Olambda__587,axiom,
    ! [Uu2: set_nat,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: code_integer] : ( aa_Code_integer_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_jd(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Cod8670293052248272184at_nat(Uua,Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_jc(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_587
tff(fact_9686_ATP_Olambda__588,axiom,
    ! [Uu2: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : ( aa_complex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_iy(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ix(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_588
tff(fact_9687_ATP_Olambda__589,axiom,
    ! [Uu2: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : ( aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_it(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_is(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_589
tff(fact_9688_ATP_Olambda__590,axiom,
    ! [Uu2: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : ( aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_ip(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_io(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_590
tff(fact_9689_ATP_Olambda__591,axiom,
    ! [Uu2: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE6480751511521678896_nat_o,Uuc: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aa_fun3621025987637128233BT_nat(aa_fun1603736613815373464BT_nat(aTP_Lamm_il(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc)),collect_nat(aa_VEB2314620471664397097_nat_o(aa_fun1998352179508271774_nat_o(aTP_Lamm_ik(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_591
tff(fact_9690_ATP_Olambda__592,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : ( aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_vn(Uu2),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,Uuc = vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uu2)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua,Uub)) ) ).

% ATP.lambda_592
tff(fact_9691_ATP_Olambda__593,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : ( aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_vm(Uu2),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_o_assn(pure_assn,Uuc = vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uu2)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua,Uub)) ) ).

% ATP.lambda_593
tff(fact_9692_ATP_Olambda__594,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_nat,Uub: fun_nat_uint32,Uuc: nat] : ( aa_nat_uint32(aa_fun7322500112117231024uint32(aa_fun6985408296456604618uint32(aTP_Lamm_rt(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_rs(Uu2),Uua),Uuc))))),aa_nat_uint32(Uub,Uuc)) ) ).

% ATP.lambda_594
tff(fact_9693_ATP_Olambda__595,axiom,
    ! [Uu2: set_literal,Uua: fun_li486744618827423812nteger,Uub: fun_Co1554742298434871494uint32,Uuc: code_integer] : ( aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(aa_fun5035814983738846737uint32(aTP_Lamm_rq(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,finite_card_literal(collect_literal(aa_Cod3435528066990957986eral_o(aa_fun5342429524804797673eral_o(aTP_Lamm_rp(Uu2),Uua),Uuc))))),aa_Cod8401701657498852871uint32(Uub,Uuc)) ) ).

% ATP.lambda_595
tff(fact_9694_ATP_Olambda__596,axiom,
    ! [Uu2: set_real,Uua: fun_re6978948838062957830nteger,Uub: fun_Co1554742298434871494uint32,Uuc: code_integer] : ( aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(aa_fun1633138777266163917uint32(aTP_Lamm_rn(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,finite_card_real(collect_real(aa_Cod1315720040876898922real_o(aa_fun4638768144124725693real_o(aTP_Lamm_rm(Uu2),Uua),Uuc))))),aa_Cod8401701657498852871uint32(Uub,Uuc)) ) ).

% ATP.lambda_596
tff(fact_9695_ATP_Olambda__597,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE9219535225822561060nteger,Uub: fun_Co1554742298434871494uint32,Uuc: code_integer] : ( aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(aa_fun1560612253975464305uint32(aTP_Lamm_rk(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_Cod5040069908416755522VEBT_o(aa_fun6317896443157543913VEBT_o(aTP_Lamm_rj(Uu2),Uua),Uuc))))),aa_Cod8401701657498852871uint32(Uub,Uuc)) ) ).

% ATP.lambda_597
tff(fact_9696_ATP_Olambda__598,axiom,
    ! [Uu2: set_literal,Uua: fun_literal_complex,Uub: fun_complex_uint32,Uuc: complex] : ( aa_complex_uint32(aa_fun947202747272686700uint32(aa_fun7541052212593818756uint32(aTP_Lamm_rh(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,finite_card_literal(collect_literal(aa_com6030452599096385139eral_o(aa_fun1482976424769362443eral_o(aTP_Lamm_rg(Uu2),Uua),Uuc))))),aa_complex_uint32(Uub,Uuc)) ) ).

% ATP.lambda_598
tff(fact_9697_ATP_Olambda__599,axiom,
    ! [Uu2: set_literal,Uua: fun_literal_nat,Uub: fun_nat_uint32,Uuc: nat] : ( aa_nat_uint32(aa_fun7322500112117231024uint32(aa_fun4475802048650532586uint32(aTP_Lamm_ra(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,finite_card_literal(collect_literal(aa_nat_fun_literal_o(aa_fun4198428272352532303eral_o(aTP_Lamm_qz(Uu2),Uua),Uuc))))),aa_nat_uint32(Uub,Uuc)) ) ).

% ATP.lambda_599
tff(fact_9698_ATP_Olambda__600,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: fun_complex_uint32,Uuc: complex] : ( aa_complex_uint32(aa_fun947202747272686700uint32(aa_fun5489866774865184576uint32(aTP_Lamm_re(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,finite_card_real(collect_real(aa_com1951324360330529881real_o(aa_fun8514437842610354715real_o(aTP_Lamm_of(Uu2),Uua),Uuc))))),aa_complex_uint32(Uub,Uuc)) ) ).

% ATP.lambda_600
tff(fact_9699_ATP_Olambda__601,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_complex_uint32,Uuc: complex] : ( aa_complex_uint32(aa_fun947202747272686700uint32(aa_fun7504237614304368612uint32(aTP_Lamm_rc(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_com3651963455866140051VEBT_o(aa_fun4726854402537413643VEBT_o(aTP_Lamm_od(Uu2),Uua),Uuc))))),aa_complex_uint32(Uub,Uuc)) ) ).

% ATP.lambda_601
tff(fact_9700_ATP_Olambda__602,axiom,
    ! [Uu2: set_real,Uua: fun_real_nat,Uub: fun_nat_uint32,Uuc: nat] : ( aa_nat_uint32(aa_fun7322500112117231024uint32(aa_fun8445541558549760422uint32(aTP_Lamm_qx(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,finite_card_real(collect_real(aa_nat_fun_real_o(aa_fun6618657382008502231real_o(aTP_Lamm_nv(Uu2),Uua),Uuc))))),aa_nat_uint32(Uub,Uuc)) ) ).

% ATP.lambda_602
tff(fact_9701_ATP_Olambda__603,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: fun_nat_uint32,Uuc: nat] : ( aa_nat_uint32(aa_fun7322500112117231024uint32(aa_fun3572208589875229258uint32(aTP_Lamm_qv(Uu2),Uua),Uub),Uuc) = aa_uint32_uint32(aa_uin3778660748068847871uint32(times_times_uint32,aa_nat_uint32(semiri2565882477558803405uint32,finite7802652506058667612T_VEBT(collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun8313996525498199887VEBT_o(aTP_Lamm_nt(Uu2),Uua),Uuc))))),aa_nat_uint32(Uub,Uuc)) ) ).

% ATP.lambda_603
tff(fact_9702_ATP_Olambda__604,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Co1900716531650368264omplex,Uub: fun_Co1900716531650368264omplex,Uuc: complex] : ( aa_complex_complex(aa_fun4919649359853996703omplex(aa_fun7424176652353795108omplex(aTP_Lamm_om(Uu2),Uua),Uub),Uuc) = groups8024822376189712711omplex(Uua,collect_Code_integer(aa_com4877492900066995176eger_o(aa_fun6433599355428082873eger_o(aTP_Lamm_ol(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_604
tff(fact_9703_ATP_Olambda__605,axiom,
    ! [Uu2: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: complex] : ( aa_complex_complex(aa_fun2120204959111876718omplex(aa_fun1822015382586975682omplex(aTP_Lamm_ok(Uu2),Uua),Uub),Uuc) = groups3049146728041665814omplex(Uua,collect_int(aa_complex_fun_int_o(aa_fun3347742875685192987_int_o(aTP_Lamm_oj(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_605
tff(fact_9704_ATP_Olambda__606,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: complex] : ( aa_complex_complex(aa_fun711518050430204434omplex(aa_fun2227483609818284298omplex(aTP_Lamm_oi(Uu2),Uua),Uub),Uuc) = groups2073611262835488442omplex(Uua,collect_nat(aa_complex_fun_nat_o(aa_fun8139692174331916755_nat_o(aTP_Lamm_oh(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_606
tff(fact_9705_ATP_Olambda__607,axiom,
    ! [Uu2: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: complex] : ( aa_complex_complex(aa_fun5897791145500046318omplex(aa_fun9191395874192406082omplex(aTP_Lamm_og(Uu2),Uua),Uub),Uuc) = groups5754745047067104278omplex(Uua,collect_real(aa_com1951324360330529881real_o(aa_fun8514437842610354715real_o(aTP_Lamm_of(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_607
tff(fact_9706_ATP_Olambda__608,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VE5569070535936029429omplex,Uub: fun_VE5569070535936029429omplex,Uuc: complex] : ( aa_complex_complex(aa_fun7854130243116081298omplex(aa_fun1272854850243878376omplex(aTP_Lamm_oe(Uu2),Uua),Uub),Uuc) = groups1794756597179926696omplex(Uua,collect_VEBT_VEBT(aa_com3651963455866140051VEBT_o(aa_fun4726854402537413643VEBT_o(aTP_Lamm_od(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_608
tff(fact_9707_ATP_Olambda__609,axiom,
    ! [Uu2: set_Code_integer,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_nat,Uuc: nat] : ( aa_nat_nat(aa_fun1707803128846032901at_nat(aa_fun2075405564639334828at_nat(aTP_Lamm_oc(Uu2),Uua),Uub),Uuc) = groups7237345082560585321er_nat(Uua,collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun4176659983681010805eger_o(aTP_Lamm_ob(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_609
tff(fact_9708_ATP_Olambda__610,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: nat] : ( aa_nat_nat(aa_fun3611238026331006550at_nat(aa_fun4158405052476944462at_nat(aTP_Lamm_oa(Uu2),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(Uua,collect_complex(aa_nat_fun_complex_o(aa_fun3931428123345706963plex_o(aTP_Lamm_nz(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_610
tff(fact_9709_ATP_Olambda__611,axiom,
    ! [Uu2: set_int,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: nat] : ( aa_nat_nat(aa_fun5507398438843482068at_nat(aa_fun3220669823809117514at_nat(aTP_Lamm_ny(Uu2),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(Uua,collect_int(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_nx(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_611
tff(fact_9710_ATP_Olambda__612,axiom,
    ! [Uu2: set_real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: nat] : ( aa_nat_nat(aa_fun3947338165632262996at_nat(aa_fun3815547947404115146at_nat(aTP_Lamm_nw(Uu2),Uua),Uub),Uuc) = groups1935376822645274424al_nat(Uua,collect_real(aa_nat_fun_real_o(aa_fun6618657382008502231real_o(aTP_Lamm_nv(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_612
tff(fact_9711_ATP_Olambda__613,axiom,
    ! [Uu2: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: fun_VEBT_VEBT_nat,Uuc: nat] : ( aa_nat_nat(aa_fun4135997101171927032at_nat(aa_fun456795606997707180at_nat(aTP_Lamm_nu(Uu2),Uua),Uub),Uuc) = groups771621172384141258BT_nat(Uua,collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun8313996525498199887VEBT_o(aTP_Lamm_nt(Uu2),Uub),Uuc))) ) ).

% ATP.lambda_613
tff(fact_9712_ATP_Olambda__614,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_int,Uub: fun_complex_complex,Uuc: int] : ( aa_int_complex(aa_fun6006737650382694510omplex(aa_fun5961093689379013058omplex(aTP_Lamm_sj(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(Uub,collect_complex(aa_int_fun_complex_o(aa_fun1485481922755881755plex_o(aTP_Lamm_si(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_614
tff(fact_9713_ATP_Olambda__615,axiom,
    ! [Uu2: set_complex,Uua: fun_co8153928105989009288nteger,Uub: fun_complex_complex,Uuc: code_integer] : ( aa_Cod5621350732091595593omplex(aa_fun7233930908441553055omplex(aa_fun2168485499150143268omplex(aTP_Lamm_sh(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(Uub,collect_complex(aa_Cod7908732550325689320plex_o(aa_fun8663194649151195321plex_o(aTP_Lamm_sg(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_615
tff(fact_9714_ATP_Olambda__616,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_int,Uub: fun_nat_real,Uuc: int] : ( aa_int_real(aa_fun4026127921582108556t_real(aa_fun4107705952538744322t_real(aTP_Lamm_sm(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(Uub,collect_nat(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_sd(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_616
tff(fact_9715_ATP_Olambda__617,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_int,Uub: fun_nat_nat,Uuc: int] : ( aa_int_nat(aa_fun6790803225114646484nt_nat(aa_fun5866268260113680714nt_nat(aTP_Lamm_se(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(Uub),collect_nat(aa_int_fun_nat_o(aa_fun6124098986320894167_nat_o(aTP_Lamm_sd(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_617
tff(fact_9716_ATP_Olambda__618,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_real,Uuc: code_integer] : ( aa_Code_integer_real(aa_fun8082968229890050365r_real(aa_fun4476568600492114404r_real(aTP_Lamm_sl(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(Uub,collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun262808356775936629_nat_o(aTP_Lamm_sb(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_618
tff(fact_9717_ATP_Olambda__619,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_Code_integer,Uub: fun_nat_nat,Uuc: code_integer] : ( aa_Code_integer_nat(aa_fun2720235278490593797er_nat(aa_fun676797813428960940er_nat(aTP_Lamm_sc(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(Uub),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun262808356775936629_nat_o(aTP_Lamm_sb(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_619
tff(fact_9718_ATP_Olambda__620,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_complex,Uub: fun_nat_real,Uuc: complex] : ( aa_complex_real(aa_fun2896631202927662350x_real(aa_fun3645564257361631750x_real(aTP_Lamm_sk(Uu2),Uua),Uub),Uuc) = groups6591440286371151544t_real(Uub,collect_nat(aa_complex_fun_nat_o(aa_fun8139692174331916755_nat_o(aTP_Lamm_oh(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_620
tff(fact_9719_ATP_Olambda__621,axiom,
    ! [Uu2: set_nat,Uua: fun_nat_complex,Uub: fun_nat_nat,Uuc: complex] : ( aa_complex_nat(aa_fun8118263755821012566ex_nat(aa_fun5276372505856632910ex_nat(aTP_Lamm_sa(Uu2),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(Uub),collect_nat(aa_complex_fun_nat_o(aa_fun8139692174331916755_nat_o(aTP_Lamm_oh(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_621
tff(fact_9720_ATP_Olambda__622,axiom,
    ! [Uu2: set_complex,Uua: fun_complex_nat,Uub: fun_complex_complex,Uuc: nat] : ( aa_nat_complex(aa_fun5031202185176517138omplex(aa_fun2450700647192873226omplex(aTP_Lamm_sf(Uu2),Uua),Uub),Uuc) = groups7754918857620584856omplex(Uub,collect_complex(aa_nat_fun_complex_o(aa_fun3931428123345706963plex_o(aTP_Lamm_nz(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_622
tff(fact_9721_ATP_Olambda__623,axiom,
    ! [Uu2: set_int,Uua: fun_int_nat,Uub: fun_int_int,Uuc: nat] : ( aa_nat_int(aa_fun5280992047749994636at_int(aa_fun4682031307293100034at_int(aTP_Lamm_sn(Uu2),Uua),Uub),Uuc) = groups4538972089207619220nt_int(Uub,collect_int(aa_nat_fun_int_o(aa_fun8596217221128839383_int_o(aTP_Lamm_nx(Uu2),Uua),Uuc))) ) ).

% ATP.lambda_623
tff(fact_9722_ATP_Olambda__624,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_ub(Uu2),Uua),Uub),Uuc) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),aa_nat_nat(suc,Uuc))) ) ).

% ATP.lambda_624
tff(fact_9723_ATP_Olambda__625,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : ( aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ua(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),aa_nat_nat(suc,Uuc))) ) ).

% ATP.lambda_625
tff(fact_9724_ATP_Olambda__626,axiom,
    ! [Uu2: fun_VE2561185598974572172i_assn,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBTi,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis6960512757483956916i_assn(aa_lis2738398937141596790i_assn(aTP_Lamm_ai(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1891944875198410415_VEBTi(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_626
tff(fact_9725_ATP_Olambda__627,axiom,
    ! [Uu2: fun_VE5577048379444288083T_assn,Uua: list_VEBT_VEBTi,Uub: list_VEBT_VEBT,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis4124290615194246587i_assn(aTP_Lamm_ag(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7265847600308530106T_VEBT(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_627
tff(fact_9726_ATP_Olambda__628,axiom,
    ! [Uu2: fun_VE1225457145571040039t_assn,Uua: list_VEBT_VEBTi,Uub: list_nat,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis9001725492116066833i_assn(aTP_Lamm_ah(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8930081998596925642Ti_nat(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_628
tff(fact_9727_ATP_Olambda__629,axiom,
    ! [Uu2: fun_VE5641251438918192003t_assn,Uua: list_VEBT_VEBTi,Uub: list_int,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis3326845930971794347i_assn(aa_lis2594938629241682541i_assn(aTP_Lamm_aj(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8927591528087875366Ti_int(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_629
tff(fact_9728_ATP_Olambda__630,axiom,
    ! [Uu2: fun_VE1571388752190420990T_assn,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis7580683478039494528i_assn(aTP_Lamm_ao(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1279224858307276611T_VEBT(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_630
tff(fact_9729_ATP_Olambda__631,axiom,
    ! [Uu2: fun_VE4931415397359990460t_assn,Uua: list_VEBT_VEBT,Uub: list_nat,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis3335532474104912652i_assn(aTP_Lamm_ap(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8296926524756676353BT_nat(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_631
tff(fact_9730_ATP_Olambda__632,axiom,
    ! [Uu2: fun_in1903134756389476099i_assn,Uua: list_int,Uub: list_VEBT_VEBTi,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis6960512757483956916i_assn(aa_lis2664486819095662317i_assn(aTP_Lamm_am(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6235239671944049190_VEBTi(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_632
tff(fact_9731_ATP_Olambda__633,axiom,
    ! [Uu2: fun_in7755481285845205148T_assn,Uua: list_int,Uub: list_VEBT_VEBT,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis2788378349880184623i_assn(aa_lis9004913286454700036i_assn(aTP_Lamm_ak(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L1664421287176695555T_VEBT(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_633
tff(fact_9732_ATP_Olambda__634,axiom,
    ! [Uu2: fun_int_fun_nat_assn,Uua: list_int,Uub: list_nat,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis510260756991402831i_assn(aa_lis5475328533724316040i_assn(aTP_Lamm_al(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L77084186935402305nt_nat(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_634
tff(fact_9733_ATP_Olambda__635,axiom,
    ! [Uu2: fun_int_fun_int_assn,Uua: list_int,Uub: list_int,Uuc: fun_VEBT_VEBTi_assn,Uud: vEBT_VEBTi] : ( aa_VEBT_VEBTi_assn(aa_fun4208009141160890610i_assn(aa_lis3326845930971794347i_assn(aa_lis8291913707704707556i_assn(aTP_Lamm_an(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L74593716426352029nt_int(Uu2,Uua,Uub)),aa_VEBT_VEBTi_assn(Uuc,Uud)) ) ).

% ATP.lambda_635
tff(fact_9734_ATP_Olambda__636,axiom,
    ! [Uu2: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: option_nat] :
      ( aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aa_set5142912993642680584t_assn(aTP_Lamm_wa(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
          aa_o_assn(pure_assn,
            ( ( Uud = some_nat(Uuc) )
            = aa_nat_o(vEBT_VEBT_min_in_set(Uua),Uuc) ))) ) ).

% ATP.lambda_636
tff(fact_9735_ATP_Olambda__637,axiom,
    ! [Uu2: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: option_nat] :
      ( aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aa_set5142912993642680584t_assn(aTP_Lamm_vw(Uu2),Uua),Uub),Uuc),Uud) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
          aa_o_assn(pure_assn,
            ( ( Uud = some_nat(Uuc) )
            = aa_nat_o(vEBT_VEBT_max_in_set(Uua),Uuc) ))) ) ).

% ATP.lambda_637
tff(fact_9736_ATP_Olambda__638,axiom,
    ! [Uu2: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: $o] : ( aa_o_assn(aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aa_set2974300032405964540o_assn(aTP_Lamm_xb(Uu2),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),aa_o_assn(pure_assn,(Uud) = aa_set_nat_o(member_nat(Uuc),Uua))) ) ).

% ATP.lambda_638
tff(fact_9737_ATP_Olambda__639,axiom,
    ! [Uu2: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : ( aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_ys(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_yr(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))))) ) ).

% ATP.lambda_639
tff(fact_9738_ATP_Olambda__640,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      ( aa_nat_VEBT_VEBT(aa_nat6729221662743847087T_VEBT(aa_nat1426624462703894328T_VEBT(aa_VEB257809287009931607T_VEBT(aa_lis5232481317877780700T_VEBT(aTP_Lamm_xp(Uu2),Uua),Uub),Uuc),Uud),Uue) = $ite(
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uud)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uuc) ),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uud),Uue)),Uu2,Uua),Uub),
          $ite(
            ( ( Uuc = Uud )
            & ( Uuc = Uue ) ),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uua),Uub),
            $let(
              xn: nat,
              xn:= 
                $ite(Uuc = Uud,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Uub))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),the_nat(vEBT_vebt_mint(Uub)))))),Uuc),
              $let(
                minn: nat,
                minn:= 
                  $ite(Uuc = Uud,xn,Uud),
                $let(
                  h: nat,
                  h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(Uua)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(Uua,h,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Uub),h),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                    $ite(
                                      xn = Uue,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                      Uue ))),Uu2,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                  $ite(xn = Uue,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,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Uue))),Uu2,newlist),Uub) ) ) ),
                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uud),Uue)),Uu2,Uua),Uub) ) ) ) ) ) ) ) ).

% ATP.lambda_640
tff(fact_9739_ATP_Olambda__641,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      ( aa_nat_option_nat(aa_nat1489030058641307643on_nat(aa_nat1972073180414880072on_nat(aa_VEB5583160467804174207on_nat(aa_lis5680555565288705232on_nat(aTP_Lamm_xw(Uu2),Uua),Uub),Uuc),Uud),Uue) = $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uud),
          some_nat(Uud),
          $let(
            l: nat,
            l:= aa_nat_nat(vEBT_VEBT_low(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(vEBT_VEBT_high(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(Uua)),
                $let(
                  maxlow: option_nat,
                  maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),h)),
                  $ite(
                    ( ( maxlow != none_nat )
                    & vEBT_VEBT_less(some_nat(l),maxlow) ),
                    aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),h),l)),
                    $let(
                      sc: option_nat,
                      sc:= vEBT_vebt_succ(Uub,h),
                      $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),the_nat(sc))))) ) ) ),
                none_nat ) ) ) ) ) ).

% ATP.lambda_641
tff(fact_9740_ATP_Olambda__642,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uua: nat,Uub: list_VEBT_VEBT,Uuc: vEBT_VEBT,Uud: nat,Uue: product_prod_nat_nat] :
      ( aa_Pro3676457433925622191T_VEBT(aa_nat6988169923125987140T_VEBT(aa_VEB5764018861059009915T_VEBT(aa_lis2657662162680668108T_VEBT(aa_nat7935216168507688033T_VEBT(aTP_Lamm_xq(Uu2),Uua),Uub),Uuc),Uud),Uue) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Uua,Uub),Uuc),aa_Pro3676457433925622191T_VEBT(produc3169358591047799142T_VEBT(aa_nat1426624462703894328T_VEBT(aa_VEB257809287009931607T_VEBT(aa_lis5232481317877780700T_VEBT(aTP_Lamm_xp(Uua),Uub),Uuc),Uud)),Uue)) ) ).

% ATP.lambda_642
tff(fact_9741_ATP_Olambda__643,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      ( aa_nat_VEBT_VEBT(aa_nat6729221662743847087T_VEBT(aa_nat1426624462703894328T_VEBT(aa_VEB257809287009931607T_VEBT(aa_lis5232481317877780700T_VEBT(aTP_Lamm_xr(Uu2),Uua),Uub),Uuc),Uud),Uue) = $let(
          xn: nat,
          xn:= 
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uud),Uud,Uuc),
          $let(
            h: nat,
            h:= aa_nat_nat(vEBT_VEBT_high(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(Uua))
              & ~ ( ( Uuc = Uud )
                  | ( Uuc = Uue ) ) ),
              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uud),Uuc,Uud)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Uue))),Uu2,list_u1324408373059187874T_VEBT(Uua,h,aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),h)),aa_nat_nat(vEBT_VEBT_low(xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),h)),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uub),h),Uub)),
              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uud),Uue)),Uu2,Uua),Uub) ) ) ) ) ).

% ATP.lambda_643
tff(fact_9742_ATP_Olambda__644,axiom,
    ! [Uu2: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: option_nat] :
      ( aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_xd(Uu2),Uua),Uub),Uuc),Uud),Uue) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
          aa_o_assn(pure_assn,
            ( ( Uue = some_nat(Uud) )
            = aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Uua),Uuc),Uud) ))) ) ).

% ATP.lambda_644
tff(fact_9743_ATP_Olambda__645,axiom,
    ! [Uu2: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: option_nat] :
      ( aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_xc(Uu2),Uua),Uub),Uuc),Uud),Uue) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
          aa_o_assn(pure_assn,
            ( ( Uue = some_nat(Uud) )
            = aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Uua),Uuc),Uud) ))) ) ).

% ATP.lambda_645
tff(fact_9744_ATP_Olambda__646,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zt(Uu2),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu2))
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),I3)),X_12)
            <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Uub),I3) ) )
        & $ite(
            Uue = Uuf,
            ! [X5: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X5),aa_lis2683774190790863254T_VEBT(set_VEBT_VEBT2,Uua))
             => ~ ? [X7: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X5),X7) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu2))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X5)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X5)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_646
tff(fact_9745_ATP_Olambda__647,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBT] : ( aa_VEB5832953974749077948T_VEBT(aTP_Lamm_nk(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_647
tff(fact_9746_ATP_Olambda__648,axiom,
    ! [Uu2: vEBT_VEBT,Uua: real] : ( aa_real_VEBT_VEBT(aTP_Lamm_nl(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_648
tff(fact_9747_ATP_Olambda__649,axiom,
    ! [Uu2: vEBT_VEBT,Uua: $o] : ( aa_o_VEBT_VEBT(aTP_Lamm_ns(Uu2),(Uua)) = Uu2 ) ).

% ATP.lambda_649
tff(fact_9748_ATP_Olambda__650,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat] : ( aa_nat_VEBT_VEBT(aTP_Lamm_nj(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_650
tff(fact_9749_ATP_Olambda__651,axiom,
    ! [Uu2: vEBT_VEBT,Uua: int] : ( aa_int_VEBT_VEBT(aTP_Lamm_nm(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_651
tff(fact_9750_ATP_Olambda__652,axiom,
    ! [Uu2: $o,Uua: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aTP_Lamm_nq,(Uu2)),Uua)
    <=> (Uu2) ) ).

% ATP.lambda_652
tff(fact_9751_ATP_Olambda__653,axiom,
    ! [Uu2: $o,Uua: int] :
      ( aa_int_o(aTP_Lamm_nr((Uu2)),Uua)
    <=> (Uu2) ) ).

% ATP.lambda_653
tff(fact_9752_ATP_Olambda__654,axiom,
    ! [Uu2: nat,Uua: vEBT_VEBT] : ( aa_VEBT_VEBT_nat(aTP_Lamm_nn(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_654
tff(fact_9753_ATP_Olambda__655,axiom,
    ! [Uu2: nat,Uua: code_integer] : ( aa_Code_integer_nat(aTP_Lamm_cy(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_655
tff(fact_9754_ATP_Olambda__656,axiom,
    ! [Uu2: nat,Uua: complex] : ( aa_complex_nat(aTP_Lamm_cx(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_656
tff(fact_9755_ATP_Olambda__657,axiom,
    ! [Uu2: nat,Uua: real] : ( aa_real_nat(aTP_Lamm_no(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_657
tff(fact_9756_ATP_Olambda__658,axiom,
    ! [Uu2: nat,Uua: $o] : ( aa_o_nat(aTP_Lamm_db(Uu2),(Uua)) = Uu2 ) ).

% ATP.lambda_658
tff(fact_9757_ATP_Olambda__659,axiom,
    ! [Uu2: nat,Uua: nat] : ( aa_nat_nat(aTP_Lamm_cz(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_659
tff(fact_9758_ATP_Olambda__660,axiom,
    ! [Uu2: nat,Uua: int] : ( aa_int_nat(aTP_Lamm_da(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_660
tff(fact_9759_ATP_Olambda__661,axiom,
    ! [Uu2: int,Uua: vEBT_VEBT] : ( aa_VEBT_VEBT_int(aTP_Lamm_np(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_661
tff(fact_9760_ATP_Olambda__662,axiom,
    ! [Uu2: int,Uua: nat] : ( aa_nat_int(aTP_Lamm_cv(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_662
tff(fact_9761_ATP_Olambda__663,axiom,
    ! [Uu2: int,Uua: int] : ( aa_int_int(aTP_Lamm_cw(Uu2),Uua) = Uu2 ) ).

% ATP.lambda_663
tff(fact_9762_ATP_Olambda__664,axiom,
    ! [Uu2: complex] : ( aa_complex_complex(aTP_Lamm_vz,Uu2) = Uu2 ) ).

% ATP.lambda_664
tff(fact_9763_ATP_Olambda__665,axiom,
    ! [Uu2: nat] : ( aa_nat_nat(aTP_Lamm_aq,Uu2) = Uu2 ) ).

% ATP.lambda_665
tff(fact_9764_ATP_Olambda__666,axiom,
    ! [Uu2: int] : ( aa_int_int(aTP_Lamm_ar,Uu2) = Uu2 ) ).

% ATP.lambda_666
tff(fact_9765_ATP_Olambda__667,axiom,
    ! [Uu2: complex] : ( aa_complex_complex(aTP_Lamm_bh,Uu2) = zero_zero_complex ) ).

% ATP.lambda_667
tff(fact_9766_ATP_Olambda__668,axiom,
    ! [Uu2: uint32] : ( aa_uint32_uint32(aTP_Lamm_hi,Uu2) = zero_zero_uint32 ) ).

% ATP.lambda_668
tff(fact_9767_ATP_Olambda__669,axiom,
    ! [Uu2: real] : ( aa_real_real(aTP_Lamm_hj,Uu2) = zero_zero_real ) ).

% ATP.lambda_669
tff(fact_9768_ATP_Olambda__670,axiom,
    ! [Uu2: rat] : ( aa_rat_rat(aTP_Lamm_hh,Uu2) = zero_zero_rat ) ).

% ATP.lambda_670
tff(fact_9769_ATP_Olambda__671,axiom,
    ! [Uu2: nat] : ( aa_nat_real(aTP_Lamm_bi,Uu2) = zero_zero_real ) ).

% ATP.lambda_671
tff(fact_9770_ATP_Olambda__672,axiom,
    ! [Uu2: nat] : ( aa_nat_nat(aTP_Lamm_bg,Uu2) = zero_zero_nat ) ).

% ATP.lambda_672
tff(fact_9771_ATP_Olambda__673,axiom,
    ! [Uu2: int] : ( aa_int_int(aTP_Lamm_bj,Uu2) = zero_zero_int ) ).

% ATP.lambda_673
tff(fact_9772_ATP_Olambda__674,axiom,
    ! [Uu2: complex] :
      ( aa_complex_o(aTP_Lamm_fg,Uu2)
    <=> $false ) ).

% ATP.lambda_674
tff(fact_9773_ATP_Olambda__675,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_wu,Uu2)
    <=> $false ) ).

% ATP.lambda_675
tff(fact_9774_ATP_Olambda__676,axiom,
    ! [Uu2: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_fh,Uu2)
    <=> $false ) ).

% ATP.lambda_676
tff(fact_9775_ATP_Olambda__677,axiom,
    ! [Uu2: $o] :
      ( aa_o_o(aTP_Lamm_fl,(Uu2))
    <=> $false ) ).

% ATP.lambda_677
tff(fact_9776_ATP_Olambda__678,axiom,
    ! [Uu2: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_fi,Uu2)
    <=> $false ) ).

% ATP.lambda_678
tff(fact_9777_ATP_Olambda__679,axiom,
    ! [Uu2: nat] :
      ( aa_nat_o(aTP_Lamm_fj,Uu2)
    <=> $false ) ).

% ATP.lambda_679
tff(fact_9778_ATP_Olambda__680,axiom,
    ! [Uu2: int] :
      ( aa_int_o(aTP_Lamm_fk,Uu2)
    <=> $false ) ).

% ATP.lambda_680
tff(fact_9779_ATP_Olambda__681,axiom,
    ! [Uu2: vEBT_VEBT] : ( aa_VEB1122092716774696213at_nat(aTP_Lamm_dc,Uu2) = suc ) ).

% ATP.lambda_681
tff(fact_9780_ATP_Olambda__682,axiom,
    ! [Uu2: real] : ( aa_real_fun_nat_nat(aTP_Lamm_dd,Uu2) = suc ) ).

% ATP.lambda_682
tff(fact_9781_ATP_Olambda__683,axiom,
    ! [Uu2: $o] : ( aa_o_fun_nat_nat(aTP_Lamm_de,(Uu2)) = suc ) ).

% ATP.lambda_683
tff(fact_9782_ATP_Olambda__684,axiom,
    ! [Uu2: nat] : ( aa_nat_fun_nat_nat(aTP_Lamm_df,Uu2) = suc ) ).

% ATP.lambda_684

% Helper facts (26)
tff(help_fconj_3_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fconj,(P2)),(Q2))
      | (Q2) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fconj,(P2)),(Q2))
      | (P2) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ (P2)
      | ~ (Q2)
      | aa_o_o(aa_o_fun_o_o(fconj,(P2)),(Q2)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fdisj,(P2)),(Q2))
      | (P2)
      | (Q2) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q2: $o,P2: $o] :
      ( ~ (Q2)
      | aa_o_o(aa_o_fun_o_o(fdisj,(P2)),(Q2)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ (P2)
      | aa_o_o(aa_o_fun_o_o(fdisj,(P2)),(Q2)) ) ).

tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X8: $o,Y: $o] :
      ( ( (X8) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X8)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X8: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X8)),(Y))
      | ( (X8) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X8: int,Y: int] :
      ( ( X8 != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X8: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X8: nat,Y: nat] :
      ( ( X8 != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X8: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X8: num,Y: num] :
      ( ( X8 != Y )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X8: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X8: complex,Y: complex] :
      ( ( X8 != Y )
      | aa_complex_o(aa_com5808801942499186519plex_o(fequal_complex,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X8: complex,Y: complex] :
      ( ~ aa_complex_o(aa_com5808801942499186519plex_o(fequal_complex,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__fun_It__Nat__Onat_M_Eo_J_T,axiom,
    ! [X8: fun_nat_o,Y: fun_nat_o] :
      ( ( X8 != Y )
      | aa_fun_nat_o_o(aa_fun4840518763754620511at_o_o(fequal_fun_nat_o,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__fun_It__Nat__Onat_M_Eo_J_T,axiom,
    ! [X8: fun_nat_o,Y: fun_nat_o] :
      ( ~ aa_fun_nat_o_o(aa_fun4840518763754620511at_o_o(fequal_fun_nat_o,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X8: set_nat,Y: set_nat] :
      ( ( X8 != Y )
      | aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X8: set_nat,Y: set_nat] :
      ( ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(fequal_set_nat,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X8: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( X8 != Y )
      | aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(fequal_VEBT_VEBT,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X8: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(fequal_VEBT_VEBT,X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X8: list_nat,Y: list_nat] :
      ( ( X8 != Y )
      | aa_list_nat_o(fequal_list_nat(X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X8: list_nat,Y: list_nat] :
      ( ~ aa_list_nat_o(fequal_list_nat(X8),Y)
      | ( X8 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_T,axiom,
    ! [X8: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( X8 != Y )
      | aa_wor4452366487969921764num1_o(aa_wor1562334678541590983num1_o(fequal882237070059529185l_num1,X8),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_T,axiom,
    ! [X8: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ~ aa_wor4452366487969921764num1_o(aa_wor1562334678541590983num1_o(fequal882237070059529185l_num1,X8),Y)
      | ( X8 = Y ) ) ).

% Conjectures (6)
tff(conj_0,hypothesis,
    s = vEBT_set_vebt(t) ).

tff(conj_1,hypothesis,
    aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,t),n) ).

tff(conj_2,hypothesis,
    aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(t),x)),r)),produc7507926704131184380et_nat(a,b)) ).

tff(conj_3,hypothesis,
    aa_set_nat_o(member_nat(xa),vEBT_set_vebt(t)) ).

tff(conj_4,hypothesis,
    ~ aa_set_nat_o(member_nat(xa),vEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(t),x))) ).

tff(conj_5,conjecture,
    xa = x ).

%------------------------------------------------------------------------------